/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 533 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QReductionProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) UsableRulesProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QReductionProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) UsableRulesProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QReductionProof [EQUIVALENT, 0 ms]
        (23) QDP
        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (25) YES
    (26) QDP
        (27) UsableRulesProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QReductionProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (32) YES
    (33) QDP
        (34) UsableRulesProof [EQUIVALENT, 0 ms]
        (35) QDP
        (36) QReductionProof [EQUIVALENT, 0 ms]
        (37) QDP
        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (39) YES
    (40) QDP
        (41) UsableRulesProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QReductionProof [EQUIVALENT, 0 ms]
        (44) QDP
        (45) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (46) YES
    (47) QDP
        (48) UsableRulesProof [EQUIVALENT, 0 ms]
        (49) QDP
        (50) QReductionProof [EQUIVALENT, 0 ms]
        (51) QDP
        (52) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (53) YES
    (54) QDP
        (55) UsableRulesProof [EQUIVALENT, 0 ms]
        (56) QDP
        (57) QReductionProof [EQUIVALENT, 0 ms]
        (58) QDP
        (59) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (60) YES
    (61) QDP
        (62) UsableRulesProof [EQUIVALENT, 0 ms]
        (63) QDP
        (64) QReductionProof [EQUIVALENT, 0 ms]
        (65) QDP
        (66) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (67) YES
    (68) QDP
        (69) UsableRulesProof [EQUIVALENT, 0 ms]
        (70) QDP
        (71) QReductionProof [EQUIVALENT, 0 ms]
        (72) QDP
        (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (74) YES
    (75) QDP
        (76) UsableRulesProof [EQUIVALENT, 0 ms]
        (77) QDP
        (78) QReductionProof [EQUIVALENT, 0 ms]
        (79) QDP
        (80) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (81) YES
    (82) QDP
        (83) UsableRulesProof [EQUIVALENT, 0 ms]
        (84) QDP
        (85) QReductionProof [EQUIVALENT, 0 ms]
        (86) QDP
        (87) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (88) YES
    (89) QDP
        (90) UsableRulesProof [EQUIVALENT, 0 ms]
        (91) QDP
        (92) QReductionProof [EQUIVALENT, 0 ms]
        (93) QDP
        (94) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (95) YES
    (96) QDP
        (97) UsableRulesProof [EQUIVALENT, 0 ms]
        (98) QDP
        (99) QReductionProof [EQUIVALENT, 0 ms]
        (100) QDP
        (101) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (102) YES
    (103) QDP
        (104) UsableRulesProof [EQUIVALENT, 0 ms]
        (105) QDP
        (106) QReductionProof [EQUIVALENT, 0 ms]
        (107) QDP
        (108) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (109) YES
    (110) QDP
        (111) UsableRulesProof [EQUIVALENT, 0 ms]
        (112) QDP
        (113) QReductionProof [EQUIVALENT, 0 ms]
        (114) QDP
        (115) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (116) YES
    (117) QDP
        (118) UsableRulesProof [EQUIVALENT, 0 ms]
        (119) QDP
        (120) QReductionProof [EQUIVALENT, 0 ms]
        (121) QDP
        (122) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (123) YES
    (124) QDP
        (125) UsableRulesProof [EQUIVALENT, 0 ms]
        (126) QDP
        (127) QReductionProof [EQUIVALENT, 0 ms]
        (128) QDP
        (129) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (130) YES
    (131) QDP
        (132) UsableRulesProof [EQUIVALENT, 0 ms]
        (133) QDP
        (134) QReductionProof [EQUIVALENT, 0 ms]
        (135) QDP
        (136) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (137) YES
    (138) QDP
        (139) UsableRulesProof [EQUIVALENT, 0 ms]
        (140) QDP
        (141) QReductionProof [EQUIVALENT, 0 ms]
        (142) QDP
        (143) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (144) YES
    (145) QDP
        (146) UsableRulesProof [EQUIVALENT, 0 ms]
        (147) QDP
        (148) QReductionProof [EQUIVALENT, 0 ms]
        (149) QDP
        (150) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (151) YES
    (152) QDP
        (153) UsableRulesProof [EQUIVALENT, 0 ms]
        (154) QDP
        (155) QReductionProof [EQUIVALENT, 0 ms]
        (156) QDP
        (157) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (158) YES
    (159) QDP
        (160) UsableRulesProof [EQUIVALENT, 0 ms]
        (161) QDP
        (162) QReductionProof [EQUIVALENT, 0 ms]
        (163) QDP
        (164) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (165) YES
    (166) QDP
        (167) UsableRulesProof [EQUIVALENT, 0 ms]
        (168) QDP
        (169) QReductionProof [EQUIVALENT, 0 ms]
        (170) QDP
        (171) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (172) YES
    (173) QDP
        (174) UsableRulesProof [EQUIVALENT, 0 ms]
        (175) QDP
        (176) QReductionProof [EQUIVALENT, 0 ms]
        (177) QDP
        (178) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (179) YES
    (180) QDP
        (181) UsableRulesProof [EQUIVALENT, 0 ms]
        (182) QDP
        (183) QReductionProof [EQUIVALENT, 0 ms]
        (184) QDP
        (185) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (186) YES
    (187) QDP
        (188) UsableRulesProof [EQUIVALENT, 0 ms]
        (189) QDP
        (190) QReductionProof [EQUIVALENT, 0 ms]
        (191) QDP
        (192) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (193) YES
    (194) QDP
        (195) UsableRulesProof [EQUIVALENT, 0 ms]
        (196) QDP
        (197) QReductionProof [EQUIVALENT, 0 ms]
        (198) QDP
        (199) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (200) YES
    (201) QDP
        (202) UsableRulesProof [EQUIVALENT, 0 ms]
        (203) QDP
        (204) QReductionProof [EQUIVALENT, 0 ms]
        (205) QDP
        (206) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (207) YES
    (208) QDP
        (209) UsableRulesProof [EQUIVALENT, 0 ms]
        (210) QDP
        (211) QReductionProof [EQUIVALENT, 0 ms]
        (212) QDP
        (213) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (214) YES
    (215) QDP
        (216) UsableRulesProof [EQUIVALENT, 0 ms]
        (217) QDP
        (218) QReductionProof [EQUIVALENT, 0 ms]
        (219) QDP
        (220) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (221) YES
    (222) QDP
        (223) UsableRulesProof [EQUIVALENT, 0 ms]
        (224) QDP
        (225) QReductionProof [EQUIVALENT, 3 ms]
        (226) QDP
        (227) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (228) YES
    (229) QDP
        (230) UsableRulesProof [EQUIVALENT, 0 ms]
        (231) QDP
        (232) QReductionProof [EQUIVALENT, 2 ms]
        (233) QDP
        (234) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (235) YES
    (236) QDP
        (237) UsableRulesProof [EQUIVALENT, 0 ms]
        (238) QDP
        (239) QReductionProof [EQUIVALENT, 0 ms]
        (240) QDP
        (241) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (242) YES
    (243) QDP
        (244) UsableRulesProof [EQUIVALENT, 0 ms]
        (245) QDP
        (246) QReductionProof [EQUIVALENT, 0 ms]
        (247) QDP
        (248) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (249) YES
    (250) QDP
        (251) UsableRulesProof [EQUIVALENT, 0 ms]
        (252) QDP
        (253) QReductionProof [EQUIVALENT, 0 ms]
        (254) QDP
        (255) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (256) YES
    (257) QDP
        (258) UsableRulesProof [EQUIVALENT, 0 ms]
        (259) QDP
        (260) QReductionProof [EQUIVALENT, 0 ms]
        (261) QDP
        (262) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (263) YES
    (264) QDP
        (265) UsableRulesProof [EQUIVALENT, 0 ms]
        (266) QDP
        (267) QDPOrderProof [EQUIVALENT, 1799 ms]
        (268) QDP
        (269) QDPOrderProof [EQUIVALENT, 1438 ms]
        (270) QDP
        (271) QDPOrderProof [EQUIVALENT, 3755 ms]
        (272) QDP
        (273) DependencyGraphProof [EQUIVALENT, 0 ms]
        (274) QDP
        (275) QDPOrderProof [EQUIVALENT, 1267 ms]
        (276) QDP
        (277) DependencyGraphProof [EQUIVALENT, 0 ms]
        (278) QDP
        (279) QDPOrderProof [EQUIVALENT, 446 ms]
        (280) QDP
        (281) DependencyGraphProof [EQUIVALENT, 0 ms]
        (282) QDP
        (283) QDPOrderProof [EQUIVALENT, 1253 ms]
        (284) QDP
        (285) DependencyGraphProof [EQUIVALENT, 0 ms]
        (286) QDP
        (287) QDPOrderProof [EQUIVALENT, 1119 ms]
        (288) QDP
        (289) QDPOrderProof [EQUIVALENT, 663 ms]
        (290) QDP
        (291) QDPOrderProof [EQUIVALENT, 2403 ms]
        (292) QDP
        (293) QDPOrderProof [EQUIVALENT, 2460 ms]
        (294) QDP
        (295) QDPOrderProof [EQUIVALENT, 326 ms]
        (296) QDP
        (297) DependencyGraphProof [EQUIVALENT, 0 ms]
        (298) QDP
        (299) QDPOrderProof [EQUIVALENT, 1310 ms]
        (300) QDP
        (301) QDPOrderProof [EQUIVALENT, 2408 ms]
        (302) QDP
        (303) QDPOrderProof [EQUIVALENT, 2458 ms]
        (304) QDP
        (305) QDPOrderProof [EQUIVALENT, 2121 ms]
        (306) QDP
        (307) QDPOrderProof [EQUIVALENT, 1046 ms]
        (308) QDP
        (309) QDPOrderProof [EQUIVALENT, 1152 ms]
        (310) QDP
        (311) QDPOrderProof [EQUIVALENT, 2090 ms]
        (312) QDP
        (313) QDPOrderProof [EQUIVALENT, 2232 ms]
        (314) QDP
        (315) QDPOrderProof [EQUIVALENT, 2126 ms]
        (316) QDP
        (317) QDPOrderProof [EQUIVALENT, 2258 ms]
        (318) QDP
        (319) QDPOrderProof [EQUIVALENT, 2425 ms]
        (320) QDP
        (321) QDPOrderProof [EQUIVALENT, 2173 ms]
        (322) QDP
        (323) DependencyGraphProof [EQUIVALENT, 0 ms]
        (324) QDP
        (325) QDPOrderProof [EQUIVALENT, 1250 ms]
        (326) QDP
        (327) DependencyGraphProof [EQUIVALENT, 0 ms]
        (328) QDP
        (329) QDPOrderProof [EQUIVALENT, 1061 ms]
        (330) QDP
        (331) QDPOrderProof [EQUIVALENT, 2049 ms]
        (332) QDP
        (333) QDPQMonotonicMRRProof [EQUIVALENT, 974 ms]
        (334) QDP
        (335) DependencyGraphProof [EQUIVALENT, 0 ms]
        (336) QDP
        (337) QDPOrderProof [EQUIVALENT, 954 ms]
        (338) QDP
        (339) QDPOrderProof [EQUIVALENT, 850 ms]
        (340) QDP
        (341) DependencyGraphProof [EQUIVALENT, 0 ms]
        (342) QDP
        (343) QDPOrderProof [EQUIVALENT, 1961 ms]
        (344) QDP
        (345) QDPOrderProof [EQUIVALENT, 2081 ms]
        (346) QDP
        (347) QDPOrderProof [EQUIVALENT, 2018 ms]
        (348) QDP
        (349) QDPQMonotonicMRRProof [EQUIVALENT, 848 ms]
        (350) QDP
        (351) QDPOrderProof [EQUIVALENT, 879 ms]
        (352) QDP
        (353) QDPOrderProof [EQUIVALENT, 2057 ms]
        (354) QDP
        (355) QDPQMonotonicMRRProof [EQUIVALENT, 778 ms]
        (356) QDP
        (357) QDPOrderProof [EQUIVALENT, 917 ms]
        (358) QDP
        (359) QDPOrderProof [EQUIVALENT, 1129 ms]
        (360) QDP
        (361) DependencyGraphProof [EQUIVALENT, 0 ms]
        (362) QDP
        (363) QDPOrderProof [EQUIVALENT, 807 ms]
        (364) QDP
        (365) QDPOrderProof [EQUIVALENT, 835 ms]
        (366) QDP
        (367) QDPOrderProof [EQUIVALENT, 1743 ms]
        (368) QDP
        (369) DependencyGraphProof [EQUIVALENT, 0 ms]
        (370) QDP
        (371) QDPOrderProof [EQUIVALENT, 931 ms]
        (372) QDP
        (373) QDPOrderProof [EQUIVALENT, 667 ms]
        (374) QDP
        (375) DependencyGraphProof [EQUIVALENT, 0 ms]
        (376) QDP
        (377) UsableRulesProof [EQUIVALENT, 0 ms]
        (378) QDP
        (379) QReductionProof [EQUIVALENT, 0 ms]
        (380) QDP
        (381) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (382) YES


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

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

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   ACTIVE(U101(tt, M, N)) -> U102^1(isNatKind(M), M, N)
   ACTIVE(U101(tt, M, N)) -> ISNATKIND(M)
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   ACTIVE(U102(tt, M, N)) -> U103^1(isNat(N), M, N)
   ACTIVE(U102(tt, M, N)) -> ISNAT(N)
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   ACTIVE(U103(tt, M, N)) -> U104^1(isNatKind(N), M, N)
   ACTIVE(U103(tt, M, N)) -> ISNATKIND(N)
   ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N))
   ACTIVE(U104(tt, M, N)) -> PLUS(x(N, M), N)
   ACTIVE(U104(tt, M, N)) -> X(N, M)
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U11(tt, V1, V2)) -> U12^1(isNatKind(V1), V1, V2)
   ACTIVE(U11(tt, V1, V2)) -> ISNATKIND(V1)
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> U13^1(isNatKind(V2), V1, V2)
   ACTIVE(U12(tt, V1, V2)) -> ISNATKIND(V2)
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   ACTIVE(U13(tt, V1, V2)) -> U14^1(isNatKind(V2), V1, V2)
   ACTIVE(U13(tt, V1, V2)) -> ISNATKIND(V2)
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   ACTIVE(U14(tt, V1, V2)) -> U15^1(isNat(V1), V2)
   ACTIVE(U14(tt, V1, V2)) -> ISNAT(V1)
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   ACTIVE(U15(tt, V2)) -> U16^1(isNat(V2))
   ACTIVE(U15(tt, V2)) -> ISNAT(V2)
   ACTIVE(U16(tt)) -> MARK(tt)
   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, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   ACTIVE(U31(tt, V1, V2)) -> U32^1(isNatKind(V1), V1, V2)
   ACTIVE(U31(tt, V1, V2)) -> ISNATKIND(V1)
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   ACTIVE(U32(tt, V1, V2)) -> U33^1(isNatKind(V2), V1, V2)
   ACTIVE(U32(tt, V1, V2)) -> ISNATKIND(V2)
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   ACTIVE(U33(tt, V1, V2)) -> U34^1(isNatKind(V2), V1, V2)
   ACTIVE(U33(tt, V1, V2)) -> ISNATKIND(V2)
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   ACTIVE(U34(tt, V1, V2)) -> U35^1(isNat(V1), V2)
   ACTIVE(U34(tt, V1, V2)) -> ISNAT(V1)
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   ACTIVE(U35(tt, V2)) -> U36^1(isNat(V2))
   ACTIVE(U35(tt, V2)) -> ISNAT(V2)
   ACTIVE(U36(tt)) -> MARK(tt)
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   ACTIVE(U41(tt, V2)) -> U42^1(isNatKind(V2))
   ACTIVE(U41(tt, V2)) -> ISNATKIND(V2)
   ACTIVE(U42(tt)) -> MARK(tt)
   ACTIVE(U51(tt)) -> MARK(tt)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   ACTIVE(U61(tt, V2)) -> U62^1(isNatKind(V2))
   ACTIVE(U61(tt, V2)) -> ISNATKIND(V2)
   ACTIVE(U62(tt)) -> MARK(tt)
   ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N))
   ACTIVE(U71(tt, N)) -> U72^1(isNatKind(N), N)
   ACTIVE(U71(tt, N)) -> ISNATKIND(N)
   ACTIVE(U72(tt, N)) -> MARK(N)
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   ACTIVE(U81(tt, M, N)) -> U82^1(isNatKind(M), M, N)
   ACTIVE(U81(tt, M, N)) -> ISNATKIND(M)
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   ACTIVE(U82(tt, M, N)) -> U83^1(isNat(N), M, N)
   ACTIVE(U82(tt, M, N)) -> ISNAT(N)
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   ACTIVE(U83(tt, M, N)) -> U84^1(isNatKind(N), M, N)
   ACTIVE(U83(tt, M, N)) -> ISNATKIND(N)
   ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M)))
   ACTIVE(U84(tt, M, N)) -> S(plus(N, M))
   ACTIVE(U84(tt, M, N)) -> PLUS(N, M)
   ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N)))
   ACTIVE(U91(tt, N)) -> U92^1(isNatKind(N))
   ACTIVE(U91(tt, N)) -> ISNATKIND(N)
   ACTIVE(U92(tt)) -> MARK(0)
   ACTIVE(isNat(0)) -> MARK(tt)
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   ACTIVE(isNat(plus(V1, V2))) -> U11^1(isNatKind(V1), V1, V2)
   ACTIVE(isNat(plus(V1, V2))) -> ISNATKIND(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(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNat(x(V1, V2))) -> U31^1(isNatKind(V1), V1, V2)
   ACTIVE(isNat(x(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatKind(0)) -> MARK(tt)
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   ACTIVE(isNatKind(plus(V1, V2))) -> U41^1(isNatKind(V1), V2)
   ACTIVE(isNatKind(plus(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> U51^1(isNatKind(V1))
   ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1)
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(x(V1, V2))) -> U61^1(isNatKind(V1), V2)
   ACTIVE(isNatKind(x(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N))
   ACTIVE(plus(N, 0)) -> U71^1(isNat(N), N)
   ACTIVE(plus(N, 0)) -> ISNAT(N)
   ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N))
   ACTIVE(plus(N, s(M))) -> U81^1(isNat(M), M, N)
   ACTIVE(plus(N, s(M))) -> ISNAT(M)
   ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N))
   ACTIVE(x(N, 0)) -> U91^1(isNat(N), N)
   ACTIVE(x(N, 0)) -> ISNAT(N)
   ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N))
   ACTIVE(x(N, s(M))) -> U101^1(isNat(M), M, N)
   ACTIVE(x(N, s(M))) -> ISNAT(M)
   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(isNat(X)) -> ACTIVE(isNat(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(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(plus(X1, X2)) -> PLUS(mark(X1), mark(X2))
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> X(mark(X1), mark(X2))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U11(X1, X2, X3)) -> U11^1(mark(X1), X2, X3)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   MARK(U12(X1, X2, X3)) -> U12^1(mark(X1), X2, X3)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   MARK(U13(X1, X2, X3)) -> U13^1(mark(X1), X2, X3)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   MARK(U14(X1, X2, X3)) -> U14^1(mark(X1), X2, X3)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   MARK(U15(X1, X2)) -> U15^1(mark(X1), X2)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U16(X)) -> ACTIVE(U16(mark(X)))
   MARK(U16(X)) -> U16^1(mark(X))
   MARK(U16(X)) -> MARK(X)
   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, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> U31^1(mark(X1), X2, X3)
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   MARK(U32(X1, X2, X3)) -> U32^1(mark(X1), X2, X3)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> U33^1(mark(X1), X2, X3)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   MARK(U34(X1, X2, X3)) -> U34^1(mark(X1), X2, X3)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   MARK(U35(X1, X2)) -> U35^1(mark(X1), X2)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U36(X)) -> ACTIVE(U36(mark(X)))
   MARK(U36(X)) -> U36^1(mark(X))
   MARK(U36(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> U41^1(mark(X1), X2)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   MARK(U42(X)) -> U42^1(mark(X))
   MARK(U42(X)) -> MARK(X)
   MARK(U51(X)) -> ACTIVE(U51(mark(X)))
   MARK(U51(X)) -> U51^1(mark(X))
   MARK(U51(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(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   MARK(U71(X1, X2)) -> U71^1(mark(X1), X2)
   MARK(U71(X1, X2)) -> MARK(X1)
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   MARK(U72(X1, X2)) -> U72^1(mark(X1), X2)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   MARK(U81(X1, X2, X3)) -> U81^1(mark(X1), X2, X3)
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   MARK(U82(X1, X2, X3)) -> U82^1(mark(X1), X2, X3)
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   MARK(U83(X1, X2, X3)) -> U83^1(mark(X1), X2, X3)
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   MARK(U84(X1, X2, X3)) -> U84^1(mark(X1), X2, X3)
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(s(X)) -> S(mark(X))
   MARK(s(X)) -> MARK(X)
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   MARK(U91(X1, X2)) -> U91^1(mark(X1), X2)
   MARK(U91(X1, X2)) -> MARK(X1)
   MARK(U92(X)) -> ACTIVE(U92(mark(X)))
   MARK(U92(X)) -> U92^1(mark(X))
   MARK(U92(X)) -> MARK(X)
   MARK(0) -> ACTIVE(0)
   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)
   ISNAT(mark(X)) -> ISNAT(X)
   ISNAT(active(X)) -> ISNAT(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)
   PLUS(mark(X1), X2) -> PLUS(X1, X2)
   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
   PLUS(active(X1), X2) -> PLUS(X1, X2)
   PLUS(X1, active(X2)) -> PLUS(X1, X2)
   X(mark(X1), X2) -> X(X1, X2)
   X(X1, mark(X2)) -> X(X1, X2)
   X(active(X1), X2) -> X(X1, X2)
   X(X1, active(X2)) -> X(X1, X2)
   U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3)
   U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3)
   U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3)
   U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3)
   U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3)
   U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3)
   U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3)
   U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3)
   U15^1(mark(X1), X2) -> U15^1(X1, X2)
   U15^1(X1, mark(X2)) -> U15^1(X1, X2)
   U15^1(active(X1), X2) -> U15^1(X1, X2)
   U15^1(X1, active(X2)) -> U15^1(X1, X2)
   U16^1(mark(X)) -> U16^1(X)
   U16^1(active(X)) -> U16^1(X)
   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, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3)
   U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3)
   U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3)
   U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3)
   U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3)
   U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3)
   U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3)
   U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3)
   U35^1(mark(X1), X2) -> U35^1(X1, X2)
   U35^1(X1, mark(X2)) -> U35^1(X1, X2)
   U35^1(active(X1), X2) -> U35^1(X1, X2)
   U35^1(X1, active(X2)) -> U35^1(X1, X2)
   U36^1(mark(X)) -> U36^1(X)
   U36^1(active(X)) -> U36^1(X)
   U41^1(mark(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, mark(X2)) -> U41^1(X1, X2)
   U41^1(active(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, active(X2)) -> U41^1(X1, X2)
   U42^1(mark(X)) -> U42^1(X)
   U42^1(active(X)) -> U42^1(X)
   U51^1(mark(X)) -> U51^1(X)
   U51^1(active(X)) -> U51^1(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(X1), X2) -> U71^1(X1, X2)
   U71^1(X1, mark(X2)) -> U71^1(X1, X2)
   U71^1(active(X1), X2) -> U71^1(X1, X2)
   U71^1(X1, active(X2)) -> U71^1(X1, X2)
   U72^1(mark(X1), X2) -> U72^1(X1, X2)
   U72^1(X1, mark(X2)) -> U72^1(X1, X2)
   U72^1(active(X1), X2) -> U72^1(X1, X2)
   U72^1(X1, active(X2)) -> U72^1(X1, X2)
   U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3)
   U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, active(X3)) -> U81^1(X1, X2, X3)
   U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3)
   U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3)
   U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3)
   U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3)
   U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3)
   U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3)
   S(mark(X)) -> S(X)
   S(active(X)) -> S(X)
   U91^1(mark(X1), X2) -> U91^1(X1, X2)
   U91^1(X1, mark(X2)) -> U91^1(X1, X2)
   U91^1(active(X1), X2) -> U91^1(X1, X2)
   U91^1(X1, active(X2)) -> U91^1(X1, X2)
   U92^1(mark(X)) -> U92^1(X)
   U92^1(active(X)) -> U92^1(X)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 38 SCCs with 114 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

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

   U92^1(active(X)) -> U92^1(X)
   U92^1(mark(X)) -> U92^1(X)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U92^1(active(X)) -> U92^1(X)
   U92^1(mark(X)) -> U92^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U92^1(active(X)) -> U92^1(X)
   U92^1(mark(X)) -> U92^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U92^1(active(X)) -> U92^1(X)
The graph contains the following edges 1 > 1


*U92^1(mark(X)) -> U92^1(X)
The graph contains the following edges 1 > 1


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

(11)
YES

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

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

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

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

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


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


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


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


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

(18)
YES

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

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

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(25)
YES

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

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

   U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3)
   U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3)
   U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3)
   U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3)
   U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3)
   U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(32)
YES

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

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

   U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3)
   U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3)
   U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3)
   U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3)
   U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3)
   U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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:
*U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(39)
YES

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

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

   U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3)
   U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3)
   U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3)
   U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3)
   U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3)
   U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(46)
YES

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

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

   U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3)
   U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, active(X3)) -> U81^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3)
   U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, active(X3)) -> U81^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3)
   U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3)
   U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3)
   U81^1(X1, X2, active(X3)) -> U81^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U81^1(X1, X2, active(X3)) -> U81^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(53)
YES

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

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

   U72^1(X1, mark(X2)) -> U72^1(X1, X2)
   U72^1(mark(X1), X2) -> U72^1(X1, X2)
   U72^1(active(X1), X2) -> U72^1(X1, X2)
   U72^1(X1, active(X2)) -> U72^1(X1, X2)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U72^1(X1, mark(X2)) -> U72^1(X1, X2)
   U72^1(mark(X1), X2) -> U72^1(X1, X2)
   U72^1(active(X1), X2) -> U72^1(X1, X2)
   U72^1(X1, active(X2)) -> U72^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U72^1(X1, mark(X2)) -> U72^1(X1, X2)
   U72^1(mark(X1), X2) -> U72^1(X1, X2)
   U72^1(active(X1), X2) -> U72^1(X1, X2)
   U72^1(X1, active(X2)) -> U72^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U72^1(X1, mark(X2)) -> U72^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U72^1(mark(X1), X2) -> U72^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U72^1(active(X1), X2) -> U72^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U72^1(X1, active(X2)) -> U72^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(60)
YES

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

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

   U71^1(X1, mark(X2)) -> U71^1(X1, X2)
   U71^1(mark(X1), X2) -> U71^1(X1, X2)
   U71^1(active(X1), X2) -> U71^1(X1, X2)
   U71^1(X1, active(X2)) -> U71^1(X1, X2)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U71^1(X1, mark(X2)) -> U71^1(X1, X2)
   U71^1(mark(X1), X2) -> U71^1(X1, X2)
   U71^1(active(X1), X2) -> U71^1(X1, X2)
   U71^1(X1, active(X2)) -> U71^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U71^1(X1, mark(X2)) -> U71^1(X1, X2)
   U71^1(mark(X1), X2) -> U71^1(X1, X2)
   U71^1(active(X1), X2) -> U71^1(X1, X2)
   U71^1(X1, active(X2)) -> U71^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U71^1(X1, mark(X2)) -> U71^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U71^1(mark(X1), X2) -> U71^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U71^1(active(X1), X2) -> U71^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U71^1(X1, active(X2)) -> U71^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(67)
YES

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

(68)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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:
*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


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

(74)
YES

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

(75)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(81)
YES

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

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

   U51^1(active(X)) -> U51^1(X)
   U51^1(mark(X)) -> U51^1(X)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U51^1(active(X)) -> U51^1(X)
   U51^1(mark(X)) -> U51^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U51^1(active(X)) -> U51^1(X)
   U51^1(mark(X)) -> U51^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U51^1(active(X)) -> U51^1(X)
The graph contains the following edges 1 > 1


*U51^1(mark(X)) -> U51^1(X)
The graph contains the following edges 1 > 1


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

(88)
YES

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

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

   U42^1(active(X)) -> U42^1(X)
   U42^1(mark(X)) -> U42^1(X)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U42^1(active(X)) -> U42^1(X)
   U42^1(mark(X)) -> U42^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U42^1(active(X)) -> U42^1(X)
   U42^1(mark(X)) -> U42^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U42^1(active(X)) -> U42^1(X)
The graph contains the following edges 1 > 1


*U42^1(mark(X)) -> U42^1(X)
The graph contains the following edges 1 > 1


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

(95)
YES

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

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

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

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

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


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


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


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


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

(102)
YES

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

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

   U36^1(active(X)) -> U36^1(X)
   U36^1(mark(X)) -> U36^1(X)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U36^1(active(X)) -> U36^1(X)
   U36^1(mark(X)) -> U36^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U36^1(active(X)) -> U36^1(X)
   U36^1(mark(X)) -> U36^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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:
*U36^1(active(X)) -> U36^1(X)
The graph contains the following edges 1 > 1


*U36^1(mark(X)) -> U36^1(X)
The graph contains the following edges 1 > 1


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

(109)
YES

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

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

   U35^1(X1, mark(X2)) -> U35^1(X1, X2)
   U35^1(mark(X1), X2) -> U35^1(X1, X2)
   U35^1(active(X1), X2) -> U35^1(X1, X2)
   U35^1(X1, active(X2)) -> U35^1(X1, X2)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U35^1(X1, mark(X2)) -> U35^1(X1, X2)
   U35^1(mark(X1), X2) -> U35^1(X1, X2)
   U35^1(active(X1), X2) -> U35^1(X1, X2)
   U35^1(X1, active(X2)) -> U35^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U35^1(X1, mark(X2)) -> U35^1(X1, X2)
   U35^1(mark(X1), X2) -> U35^1(X1, X2)
   U35^1(active(X1), X2) -> U35^1(X1, X2)
   U35^1(X1, active(X2)) -> U35^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U35^1(X1, mark(X2)) -> U35^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U35^1(mark(X1), X2) -> U35^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U35^1(active(X1), X2) -> U35^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U35^1(X1, active(X2)) -> U35^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(116)
YES

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

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

   U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3)
   U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3)
   U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3)
   U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3)
   U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3)
   U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(123)
YES

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

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

   U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3)
   U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3)
   U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3)
   U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3)
   U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3)
   U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(130)
YES

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

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

   U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3)
   U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3)
   U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3)
   U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3)
   U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3)
   U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(137)
YES

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

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

   U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3)
   U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3)
   U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3)
   U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3)
   U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3)
   U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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:
*U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(144)
YES

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

(145)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(151)
YES

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

(152)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(158)
YES

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

(159)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(165)
YES

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

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

   U16^1(active(X)) -> U16^1(X)
   U16^1(mark(X)) -> U16^1(X)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U16^1(active(X)) -> U16^1(X)
   U16^1(mark(X)) -> U16^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U16^1(active(X)) -> U16^1(X)
   U16^1(mark(X)) -> U16^1(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U16^1(active(X)) -> U16^1(X)
The graph contains the following edges 1 > 1


*U16^1(mark(X)) -> U16^1(X)
The graph contains the following edges 1 > 1


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

(172)
YES

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

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

   U15^1(X1, mark(X2)) -> U15^1(X1, X2)
   U15^1(mark(X1), X2) -> U15^1(X1, X2)
   U15^1(active(X1), X2) -> U15^1(X1, X2)
   U15^1(X1, active(X2)) -> U15^1(X1, X2)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U15^1(X1, mark(X2)) -> U15^1(X1, X2)
   U15^1(mark(X1), X2) -> U15^1(X1, X2)
   U15^1(active(X1), X2) -> U15^1(X1, X2)
   U15^1(X1, active(X2)) -> U15^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U15^1(X1, mark(X2)) -> U15^1(X1, X2)
   U15^1(mark(X1), X2) -> U15^1(X1, X2)
   U15^1(active(X1), X2) -> U15^1(X1, X2)
   U15^1(X1, active(X2)) -> U15^1(X1, X2)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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:
*U15^1(X1, mark(X2)) -> U15^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U15^1(mark(X1), X2) -> U15^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U15^1(active(X1), X2) -> U15^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U15^1(X1, active(X2)) -> U15^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(179)
YES

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

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

   U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3)
   U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3)
   U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3)
   U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3)
   U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3)
   U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(186)
YES

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

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

   U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3)
   U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3)
   U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3)
   U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3)
   U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3)
   U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(193)
YES

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

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

   U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3)
   U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3)
   U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3)
   U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3)
   U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3)
   U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(200)
YES

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

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

   U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3)
   U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

   U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3)
   U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

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

   U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3)
   U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3)
   U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3)
   U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(207)
YES

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

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

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

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

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

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(212)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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


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


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


*X(X1, active(X2)) -> X(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:

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

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(217)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(219)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

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


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


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


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


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

(221)
YES

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

(222)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(224)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3)
   U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(226)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3)
   U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(228)
YES

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

(229)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(231)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(233)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(235)
YES

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

(236)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(238)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3)
   U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(240)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3)
   U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(242)
YES

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

(243)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(245)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATKIND(active(X)) -> ISNATKIND(X)
   ISNATKIND(mark(X)) -> ISNATKIND(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(247)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATKIND(active(X)) -> ISNATKIND(X)
   ISNATKIND(mark(X)) -> ISNATKIND(X)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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:
*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


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

(249)
YES

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

(250)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(252)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3)
   U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(254)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3)
   U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(256)
YES

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

(257)
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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(259)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3)
   U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(261)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3)
   U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)

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

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

From the DPs we obtained the following set of size-change graphs:
*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


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

(263)
YES

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

(264)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> ACTIVE(U16(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> ACTIVE(U36(mark(X)))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, N)) -> MARK(N)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N)))
   MARK(U92(X)) -> ACTIVE(U92(mark(X)))
   ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N))
   MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N))
   MARK(U91(X1, X2)) -> MARK(X1)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U16(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U36(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(U51(X)) -> ACTIVE(U51(mark(X)))
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(U92(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   active(U16(tt)) -> mark(tt)
   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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   active(U36(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   active(U72(tt, N)) -> mark(N)
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   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(isNat(X)) -> active(isNat(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   mark(U16(X)) -> active(U16(mark(X)))
   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, X3)) -> active(U31(mark(X1), X2, X3))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   mark(U36(X)) -> active(U36(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(mark(X)))
   mark(U51(X)) -> active(U51(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   mark(U92(X)) -> active(U92(mark(X)))
   mark(0) -> active(0)
   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)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(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)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(mark(X)) -> U16(X)
   U16(active(X)) -> U16(X)
   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, X3) -> U31(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(mark(X)) -> U36(X)
   U36(active(X)) -> U36(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U51(mark(X)) -> U51(X)
   U51(active(X)) -> U51(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(X1), X2) -> U71(X1, X2)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U92(mark(X)) -> U92(X)
   U92(active(X)) -> U92(X)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(266)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> ACTIVE(U16(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> ACTIVE(U36(mark(X)))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, N)) -> MARK(N)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N)))
   MARK(U92(X)) -> ACTIVE(U92(mark(X)))
   ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N))
   MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N))
   MARK(U91(X1, X2)) -> MARK(X1)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U16(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U36(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(U51(X)) -> ACTIVE(U51(mark(X)))
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(U92(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U16(X)) -> ACTIVE(U16(mark(X)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   MARK(U36(X)) -> ACTIVE(U36(mark(X)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(U92(X)) -> ACTIVE(U92(mark(X)))
   MARK(U51(X)) -> ACTIVE(U51(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( U11_3(x_1, ..., x_3) ) = 2
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = 2
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U15_2(x_1, x_2) ) = 2
POL( U16_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = 0
POL( U31_3(x_1, ..., x_3) ) = 2
POL( U32_3(x_1, ..., x_3) ) = 2
POL( U33_3(x_1, ..., x_3) ) = 2
POL( U34_3(x_1, ..., x_3) ) = 2
POL( U35_2(x_1, x_2) ) = 2
POL( U36_1(x_1) ) = 0
POL( U41_2(x_1, x_2) ) = 2
POL( U42_1(x_1) ) = 2
POL( U51_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = 2
POL( U72_2(x_1, x_2) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U91_2(x_1, x_2) ) = 2
POL( U92_1(x_1) ) = max{0, -2}
POL( plus_2(x_1, x_2) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( x_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 2

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

   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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)


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

(268)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, N)) -> MARK(N)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M)))
   ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N)))
   ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N))
   MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N))
   MARK(U91(X1, X2)) -> MARK(X1)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U16(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U36(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(U92(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U11_3(x_1, ..., x_3) ) = 2
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = 2
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U15_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_3(x_1, ..., x_3) ) = 2
POL( U32_3(x_1, ..., x_3) ) = 2
POL( U33_3(x_1, ..., x_3) ) = 2
POL( U34_3(x_1, ..., x_3) ) = 2
POL( U35_2(x_1, x_2) ) = 2
POL( U41_2(x_1, x_2) ) = 2
POL( U42_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_2(x_1, x_2) ) = 2
POL( U72_2(x_1, x_2) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U91_2(x_1, x_2) ) = 2
POL( plus_2(x_1, x_2) ) = 2
POL( x_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = 2x_1 + 2
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U16_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U36_1(x_1) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U92_1(x_1) ) = 0
POL( U51_1(x_1) ) = max{0, -2}
POL( 0 ) = 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)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(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)


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

(270)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, N)) -> MARK(N)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M)))
   ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N)))
   ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N))
   MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N))
   MARK(U91(X1, X2)) -> MARK(X1)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U16(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U36(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(U92(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(271) 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, M, N)) -> MARK(plus(x(N, M), N))
   ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N))
   ACTIVE(U72(tt, N)) -> MARK(N)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M)))
   ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N))
   ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N))
   ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N))
   ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N))
   MARK(U91(X1, X2)) -> MARK(X1)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   MARK(U71(X1, X2)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(U92(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
MARK(x1)  =  x1

U102(x1, x2, x3)  =  U102(x1, x2, x3)

ACTIVE(x1)  =  x1

mark(x1)  =  x1

U101(x1, x2, x3)  =  U101(x1, x2, x3)

tt  =  tt

isNatKind(x1)  =  isNatKind

U103(x1, x2, x3)  =  U103(x1, x2, x3)

isNat(x1)  =  isNat

U104(x1, x2, x3)  =  U104(x1, x2, x3)

plus(x1, x2)  =  plus(x1, x2)

x(x1, x2)  =  x(x1, x2)

U11(x1, x2, x3)  =  x1

U12(x1, x2, x3)  =  x1

U13(x1, x2, x3)  =  x1

U14(x1, x2, x3)  =  x1

U15(x1, x2)  =  x1

U16(x1)  =  x1

U21(x1, x2)  =  x1

U22(x1, x2)  =  x1

U23(x1)  =  x1

U31(x1, x2, x3)  =  x1

U32(x1, x2, x3)  =  x1

U33(x1, x2, x3)  =  x1

U34(x1, x2, x3)  =  x1

U35(x1, x2)  =  x1

U36(x1)  =  x1

U41(x1, x2)  =  x1

U42(x1)  =  x1

U61(x1, x2)  =  x1

U62(x1)  =  x1

U71(x1, x2)  =  U71(x1, x2)

U72(x1, x2)  =  U72(x1, x2)

U81(x1, x2, x3)  =  U81(x1, x2, x3)

U82(x1, x2, x3)  =  U82(x1, x2, x3)

U83(x1, x2, x3)  =  U83(x1, x2, x3)

U84(x1, x2, x3)  =  U84(x1, x2, x3)

s(x1)  =  s(x1)

U91(x1, x2)  =  U91(x1)

U92(x1)  =  U92(x1)

0  =  0

U51(x1)  =  x1

active(x1)  =  x1


Recursive path order with status [RPO].
Quasi-Precedence: [U102_3, U101_3, U103_3, U104_3, x_2] > [tt, isNatKind, isNat, plus_2, U81_3, U82_3, U83_3, U84_3, U91_1, U92_1, 0] > U71_2 > U72_2
[U102_3, U101_3, U103_3, U104_3, x_2] > [tt, isNatKind, isNat, plus_2, U81_3, U82_3, U83_3, U84_3, U91_1, U92_1, 0] > s_1

Status: U102_3: [2,3,1]
U101_3: [2,3,1]
tt: multiset status
isNatKind: []
U103_3: [2,3,1]
isNat: []
U104_3: [2,3,1]
plus_2: [2,1]
x_2: [2,1]
U71_2: [2,1]
U72_2: multiset status
U81_3: [2,3,1]
U82_3: [2,3,1]
U83_3: [2,3,1]
U84_3: [2,3,1]
s_1: multiset status
U91_1: multiset status
U92_1: multiset status
0: multiset status


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)


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

(272)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N)))
   MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U16(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U36(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

We have to consider all minimal (P,Q,R)-chains.
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(273) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
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(274)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, -2}
POL( U11_3(x_1, ..., x_3) ) = 2
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = 2
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U15_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_3(x_1, ..., x_3) ) = 2
POL( U32_3(x_1, ..., x_3) ) = 2
POL( U33_3(x_1, ..., x_3) ) = 2
POL( U34_3(x_1, ..., x_3) ) = 2
POL( U35_2(x_1, x_2) ) = 2
POL( U41_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U71_2(x_1, x_2) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 0
POL( U91_2(x_1, x_2) ) = 2
POL( plus_2(x_1, x_2) ) = 0
POL( x_2(x_1, x_2) ) = 0
POL( mark_1(x_1) ) = 2x_1 + 2
POL( active_1(x_1) ) = max{0, -2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U16_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( U36_1(x_1) ) = max{0, -2}
POL( U42_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U92_1(x_1) ) = max{0, -2}
POL( U51_1(x_1) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 2

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

   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)


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

(276)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N))
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(277) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(278)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(279) 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(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   MARK(U91(X1, X2)) -> ACTIVE(U91(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( U101_3(x_1, ..., x_3) ) = 1
POL( U102_3(x_1, ..., x_3) ) = 1
POL( U103_3(x_1, ..., x_3) ) = max{0, -2}
POL( U11_3(x_1, ..., x_3) ) = 1
POL( U12_3(x_1, ..., x_3) ) = 1
POL( U13_3(x_1, ..., x_3) ) = 1
POL( U14_3(x_1, ..., x_3) ) = 1
POL( U15_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U31_3(x_1, ..., x_3) ) = 1
POL( U32_3(x_1, ..., x_3) ) = 1
POL( U33_3(x_1, ..., x_3) ) = 1
POL( U34_3(x_1, ..., x_3) ) = 1
POL( U35_2(x_1, x_2) ) = 1
POL( U41_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( U81_3(x_1, ..., x_3) ) = 1
POL( U82_3(x_1, ..., x_3) ) = 1
POL( U83_3(x_1, ..., x_3) ) = 0
POL( U91_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( tt ) = 2
POL( isNatKind_1(x_1) ) = 1
POL( isNat_1(x_1) ) = 1
POL( U104_3(x_1, ..., x_3) ) = 2
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U16_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2
POL( U36_1(x_1) ) = max{0, x_1 - 2}
POL( U42_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U72_2(x_1, x_2) ) = 2x_1 + 2
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U92_1(x_1) ) = 2
POL( U51_1(x_1) ) = 2
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 1

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

   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)


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

(280)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(281) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(282)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(283) 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(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 0
POL( U11_3(x_1, ..., x_3) ) = 2
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = 2
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U15_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_3(x_1, ..., x_3) ) = 2
POL( U32_3(x_1, ..., x_3) ) = 2
POL( U33_3(x_1, ..., x_3) ) = 2
POL( U34_3(x_1, ..., x_3) ) = 2
POL( U35_2(x_1, x_2) ) = 2
POL( U41_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( active_1(x_1) ) = x_1 + 2
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 1}
POL( isNat_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 0
POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( x_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U16_1(x_1) ) = 2x_1 + 2
POL( U23_1(x_1) ) = max{0, -2}
POL( U36_1(x_1) ) = max{0, x_1 - 2}
POL( U42_1(x_1) ) = 1
POL( U62_1(x_1) ) = x_1 + 2
POL( U71_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U72_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U83_3(x_1, ..., x_3) ) = 2x_2 + 2x_3
POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( s_1(x_1) ) = x_1 + 2
POL( U91_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U92_1(x_1) ) = 2
POL( U51_1(x_1) ) = max{0, x_1 - 2}
POL( 0 ) = 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)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)


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

(284)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(285) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(286)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(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( U101_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 1
POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U14_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U15_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U34_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U35_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2
POL( mark_1(x_1) ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 1}
POL( active_1(x_1) ) = max{0, x_1 - 1}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = x_2 + 2
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 1}
POL( x_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U16_1(x_1) ) = x_1 + 1
POL( U23_1(x_1) ) = x_1 + 1
POL( U36_1(x_1) ) = 2x_1 + 1
POL( U42_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = 2x_1 + 1
POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U72_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U91_2(x_1, x_2) ) = x_1 + 2
POL( U92_1(x_1) ) = 0
POL( U51_1(x_1) ) = x_1 + 1
POL( 0 ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(288)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(289) 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) ) = x_1 + 1
POL( U101_3(x_1, ..., x_3) ) = 0
POL( U11_3(x_1, ..., x_3) ) = 1
POL( U12_3(x_1, ..., x_3) ) = 1
POL( U13_3(x_1, ..., x_3) ) = 1
POL( U14_3(x_1, ..., x_3) ) = 1
POL( U15_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U31_3(x_1, ..., x_3) ) = 1
POL( U32_3(x_1, ..., x_3) ) = 1
POL( U33_3(x_1, ..., x_3) ) = 1
POL( U34_3(x_1, ..., x_3) ) = 1
POL( U35_2(x_1, x_2) ) = 1
POL( U41_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( mark_1(x_1) ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 1
POL( U103_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( isNat_1(x_1) ) = 1
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 1}
POL( plus_2(x_1, x_2) ) = max{0, x_2 - 1}
POL( x_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U16_1(x_1) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = 2
POL( U36_1(x_1) ) = max{0, 2x_1 - 2}
POL( U42_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U72_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U82_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U83_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 1}
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U91_2(x_1, x_2) ) = x_2 + 2
POL( U92_1(x_1) ) = 2
POL( U51_1(x_1) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 2

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(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)


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

(290)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U61(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(292)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(293) 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(isNatKind(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(294)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(295) 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, 2x_1 - 1}
POL( U11_3(x_1, ..., x_3) ) = 1
POL( U12_3(x_1, ..., x_3) ) = 1
POL( U13_3(x_1, ..., x_3) ) = 1
POL( U14_3(x_1, ..., x_3) ) = 1
POL( U15_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U31_3(x_1, ..., x_3) ) = 1
POL( U32_3(x_1, ..., x_3) ) = 1
POL( U33_3(x_1, ..., x_3) ) = 1
POL( U34_3(x_1, ..., x_3) ) = 1
POL( U35_2(x_1, x_2) ) = 1
POL( U41_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( active_1(x_1) ) = x_1 + 2
POL( U101_3(x_1, ..., x_3) ) = x_2 + 2
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 1
POL( U103_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( isNat_1(x_1) ) = 1
POL( U104_3(x_1, ..., x_3) ) = 2
POL( plus_2(x_1, x_2) ) = 2
POL( x_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U16_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U36_1(x_1) ) = max{0, x_1 - 2}
POL( U42_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = 2
POL( U72_2(x_1, x_2) ) = x_1 + 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( U84_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( s_1(x_1) ) = 2
POL( U91_2(x_1, x_2) ) = x_2 + 2
POL( U92_1(x_1) ) = 0
POL( U51_1(x_1) ) = max{0, -2}
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 1

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(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)


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

(296)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))
   ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(298)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   MARK(U62(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(299) 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( U11_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U14_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U15_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U34_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U35_2(x_1, x_2) ) = x_1 + 1
POL( U41_2(x_1, x_2) ) = x_1 + 1
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( active_1(x_1) ) = 2x_1 + 2
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( x_2(x_1, x_2) ) = 0
POL( U16_1(x_1) ) = x_1 + 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U36_1(x_1) ) = 2x_1 + 1
POL( U42_1(x_1) ) = x_1 + 1
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = x_1 + 2
POL( U71_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U82_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = x_3 + 2
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U92_1(x_1) ) = 2
POL( U51_1(x_1) ) = x_1 + 1
POL( 0 ) = 2
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(300)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U31(X1, X2, X3)) -> MARK(X1)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U31(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(302)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U32(X1, X2, X3)) -> MARK(X1)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U32(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

   <<<
 POL(U62(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(304)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
   MARK(U36(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U71(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(306)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
   MARK(U36(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U11_3(x_1, ..., x_3) ) = 2
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = 2
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U15_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_3(x_1, ..., x_3) ) = 2
POL( U32_3(x_1, ..., x_3) ) = 2
POL( U33_3(x_1, ..., x_3) ) = 2
POL( U34_3(x_1, ..., x_3) ) = 2
POL( U35_2(x_1, x_2) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( U102_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( active_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( isNat_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( x_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U16_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U36_1(x_1) ) = max{0, 2x_1 - 2}
POL( U42_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 1}
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( U72_2(x_1, x_2) ) = 2x_1 + 2
POL( U81_3(x_1, ..., x_3) ) = x_2 + x_3 + 1
POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U83_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U84_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U92_1(x_1) ) = max{0, 2x_1 - 2}
POL( U51_1(x_1) ) = max{0, 2x_1 - 2}
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 2

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)


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

(308)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U36(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U36(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_3(x_1, ..., x_3) ) = x_1 + 1
POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U14_3(x_1, ..., x_3) ) = x_1 + 1
POL( U15_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = max{0, -2}
POL( U32_3(x_1, ..., x_3) ) = 1
POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U34_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U41_2(x_1, x_2) ) = 2x_1 + 1
POL( mark_1(x_1) ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 1
POL( active_1(x_1) ) = x_1 + 1
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U16_1(x_1) ) = x_1 + 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U35_2(x_1, x_2) ) = 2x_1 + 1
POL( U36_1(x_1) ) = 2x_1 + 2
POL( U42_1(x_1) ) = x_1 + 1
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 1
POL( U71_2(x_1, x_2) ) = max{0, 2x_2 - 1}
POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = x_3 + 2
POL( U84_3(x_1, ..., x_3) ) = x_2 + 2
POL( s_1(x_1) ) = 0
POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U92_1(x_1) ) = 1
POL( U51_1(x_1) ) = 2x_1 + 1
POL( 0 ) = 1
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(310)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U13(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U13(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(312)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U12(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

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

   <<<
 POL(plus(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

   <<<
 POL(U16(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(314)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U34(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U34(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

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

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(x(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U62(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(316)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(317) 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)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

   <<<
 POL(mark(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

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

   <<<
 POL(U62(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[2A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(318)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2, X3)) -> MARK(X1)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

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

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(x(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(320)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U35(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U35(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(322)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(324)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 1}
POL( U11_3(x_1, ..., x_3) ) = 1
POL( U12_3(x_1, ..., x_3) ) = 1
POL( U13_3(x_1, ..., x_3) ) = 1
POL( U14_3(x_1, ..., x_3) ) = 1
POL( U15_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U31_3(x_1, ..., x_3) ) = 1
POL( U32_3(x_1, ..., x_3) ) = 1
POL( U33_3(x_1, ..., x_3) ) = 1
POL( U34_3(x_1, ..., x_3) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = 1
POL( mark_1(x_1) ) = max{0, -2}
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( active_1(x_1) ) = max{0, -2}
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 1
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 1
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( x_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U16_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( U35_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U36_1(x_1) ) = 1
POL( U42_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2x_1 + 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( U72_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U92_1(x_1) ) = 2
POL( U51_1(x_1) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 1

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)


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

(326)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(328)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U33(X1, X2, X3)) -> ACTIVE(U33(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( U11_3(x_1, ..., x_3) ) = 2
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = 2
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U15_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_3(x_1, ..., x_3) ) = 2
POL( U32_3(x_1, ..., x_3) ) = 2
POL( U33_3(x_1, ..., x_3) ) = 0
POL( U41_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( U101_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( isNat_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( x_2(x_1, x_2) ) = max{0, -2}
POL( U16_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U35_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U36_1(x_1) ) = 2
POL( U42_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = 2x_2 + 2
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = x_2 + 2
POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U84_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U91_2(x_1, x_2) ) = 2x_2 + 2
POL( U92_1(x_1) ) = max{0, 2x_1 - 2}
POL( U51_1(x_1) ) = max{0, -2}
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 1

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)


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

(330)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

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

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(X1, X2, X3)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(332)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(333) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2
   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(U11(x_1, x_2, x_3)) = 1
   POL(U12(x_1, x_2, x_3)) = 1
   POL(U13(x_1, x_2, x_3)) = 1
   POL(U14(x_1, x_2, x_3)) = 1
   POL(U15(x_1, x_2)) = 1
   POL(U16(x_1)) = 0
   POL(U21(x_1, x_2)) = 1
   POL(U22(x_1, x_2)) = 1
   POL(U23(x_1)) = 0
   POL(U31(x_1, x_2, x_3)) = 1
   POL(U32(x_1, x_2, x_3)) = 0
   POL(U33(x_1, x_2, x_3)) = 0
   POL(U34(x_1, x_2, x_3)) = 0
   POL(U35(x_1, x_2)) = 0
   POL(U36(x_1)) = 0
   POL(U41(x_1, x_2)) = 1
   POL(U42(x_1)) = 0
   POL(U51(x_1)) = 0
   POL(U61(x_1, x_2)) = 0
   POL(U62(x_1)) = 0
   POL(U71(x_1, x_2)) = 0
   POL(U72(x_1, x_2)) = 0
   POL(U81(x_1, x_2, x_3)) = 0
   POL(U82(x_1, x_2, x_3)) = 0
   POL(U83(x_1, x_2, x_3)) = 0
   POL(U84(x_1, x_2, x_3)) = 0
   POL(U91(x_1, x_2)) = 0
   POL(U92(x_1)) = 0
   POL(active(x_1)) = 0
   POL(isNat(x_1)) = 1
   POL(isNatKind(x_1)) = 1
   POL(mark(x_1)) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tt) = 0
   POL(x(x_1, x_2)) = 0


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

(334)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(336)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U33(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U33(X1, X2, X3)) -> 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( U11_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U13_3(x_1, ..., x_3) ) = max{0, -2}
POL( U14_3(x_1, ..., x_3) ) = x_1 + 1
POL( U15_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = 2x_1 + 1
POL( mark_1(x_1) ) = max{0, -2}
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( active_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 1
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( plus_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( x_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U16_1(x_1) ) = x_1 + 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U34_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U35_2(x_1, x_2) ) = 2x_1 + 2
POL( U36_1(x_1) ) = 2
POL( U42_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_2(x_1, x_2) ) = x_2 + 2
POL( U72_2(x_1, x_2) ) = x_2 + 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( s_1(x_1) ) = x_1 + 2
POL( U91_2(x_1, x_2) ) = max{0, -2}
POL( U92_1(x_1) ) = 1
POL( U51_1(x_1) ) = 2x_1 + 1
POL( 0 ) = 2
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(338)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U11_3(x_1, ..., x_3) ) = 2
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = 2
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U15_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_3(x_1, ..., x_3) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( isNat_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( plus_2(x_1, x_2) ) = max{0, -2}
POL( x_2(x_1, x_2) ) = 2
POL( U16_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 2
POL( U32_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U33_3(x_1, ..., x_3) ) = x_1 + 2
POL( U34_3(x_1, ..., x_3) ) = x_1 + x_3 + 2
POL( U35_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U36_1(x_1) ) = 0
POL( U42_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 0
POL( U71_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U72_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U82_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U83_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2
POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U92_1(x_1) ) = 0
POL( U51_1(x_1) ) = 2
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 2

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)


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

(340)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(342)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U22(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(344)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U14(X1, X2, X3)) -> MARK(X1)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U14(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

   <<<
 POL(U16(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

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

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U101(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(346)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[1A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U16(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U62(x_1)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U91(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U92(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(348)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(349) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2
   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(U11(x_1, x_2, x_3)) = 1
   POL(U12(x_1, x_2, x_3)) = 1
   POL(U13(x_1, x_2, x_3)) = 1
   POL(U14(x_1, x_2, x_3)) = 1
   POL(U15(x_1, x_2)) = 0
   POL(U16(x_1)) = 0
   POL(U21(x_1, x_2)) = 1
   POL(U22(x_1, x_2)) = 1
   POL(U23(x_1)) = 0
   POL(U31(x_1, x_2, x_3)) = 0
   POL(U32(x_1, x_2, x_3)) = 0
   POL(U33(x_1, x_2, x_3)) = 0
   POL(U34(x_1, x_2, x_3)) = 0
   POL(U35(x_1, x_2)) = 0
   POL(U36(x_1)) = 0
   POL(U41(x_1, x_2)) = 1
   POL(U42(x_1)) = 0
   POL(U51(x_1)) = 0
   POL(U61(x_1, x_2)) = 0
   POL(U62(x_1)) = 0
   POL(U71(x_1, x_2)) = 0
   POL(U72(x_1, x_2)) = 0
   POL(U81(x_1, x_2, x_3)) = 0
   POL(U82(x_1, x_2, x_3)) = 0
   POL(U83(x_1, x_2, x_3)) = 0
   POL(U84(x_1, x_2, x_3)) = 0
   POL(U91(x_1, x_2)) = 0
   POL(U92(x_1)) = 0
   POL(active(x_1)) = 0
   POL(isNat(x_1)) = 1
   POL(isNatKind(x_1)) = 1
   POL(mark(x_1)) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tt) = 0
   POL(x(x_1, x_2)) = 0


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

(350)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U16(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U16(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_3(x_1, ..., x_3) ) = max{0, -2}
POL( U13_3(x_1, ..., x_3) ) = max{0, -2}
POL( U14_3(x_1, ..., x_3) ) = 1
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = 0
POL( U41_2(x_1, x_2) ) = 2x_1 + 1
POL( mark_1(x_1) ) = max{0, -2}
POL( U102_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( active_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( plus_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( x_2(x_1, x_2) ) = 2
POL( U15_2(x_1, x_2) ) = x_1 + 1
POL( U16_1(x_1) ) = 2x_1 + 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = 2
POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U33_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U35_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U36_1(x_1) ) = 2
POL( U42_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, -2}
POL( U71_2(x_1, x_2) ) = 2
POL( U72_2(x_1, x_2) ) = 2
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U82_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U83_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U84_3(x_1, ..., x_3) ) = x_1 + 2
POL( s_1(x_1) ) = 2
POL( U91_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U92_1(x_1) ) = 2
POL( U51_1(x_1) ) = 2x_1 + 1
POL( 0 ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(352)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0A]]
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U13(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U14(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U15(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

   <<<
 POL(U21(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[5A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U101(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[5A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3
>>>

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[5A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[5A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[2A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[5A]] 	* 	x_2
>>>

   <<<
 POL(U16(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[5A]] 	* 	x_3
>>>

   <<<
 POL(U32(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[5A]] 	* 	x_3
>>>

   <<<
 POL(U33(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[5A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3
>>>

   <<<
 POL(U34(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U35(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U36(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

   <<<
 POL(U71(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U81(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U82(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U83(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U84(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

   <<<
 POL(U92(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


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

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   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(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(354)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U42(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(355) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2
   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(U11(x_1, x_2, x_3)) = 1
   POL(U12(x_1, x_2, x_3)) = 1
   POL(U13(x_1, x_2, x_3)) = 1
   POL(U14(x_1, x_2, x_3)) = 1
   POL(U15(x_1, x_2)) = 0
   POL(U16(x_1)) = 0
   POL(U21(x_1, x_2)) = 1
   POL(U22(x_1, x_2)) = 1
   POL(U23(x_1)) = 0
   POL(U31(x_1, x_2, x_3)) = 0
   POL(U32(x_1, x_2, x_3)) = 0
   POL(U33(x_1, x_2, x_3)) = 0
   POL(U34(x_1, x_2, x_3)) = 0
   POL(U35(x_1, x_2)) = 0
   POL(U36(x_1)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(U42(x_1)) = 0
   POL(U51(x_1)) = 0
   POL(U61(x_1, x_2)) = 0
   POL(U62(x_1)) = 0
   POL(U71(x_1, x_2)) = 0
   POL(U72(x_1, x_2)) = 0
   POL(U81(x_1, x_2, x_3)) = 0
   POL(U82(x_1, x_2, x_3)) = 0
   POL(U83(x_1, x_2, x_3)) = 0
   POL(U84(x_1, x_2, x_3)) = 0
   POL(U91(x_1, x_2)) = 0
   POL(U92(x_1)) = 0
   POL(active(x_1)) = 0
   POL(isNat(x_1)) = 1
   POL(isNatKind(x_1)) = 1
   POL(mark(x_1)) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tt) = 0
   POL(x(x_1, x_2)) = 0


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

(356)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U42(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U42(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_3(x_1, ..., x_3) ) = 1
POL( U12_3(x_1, ..., x_3) ) = max{0, -2}
POL( U13_3(x_1, ..., x_3) ) = max{0, -2}
POL( U14_3(x_1, ..., x_3) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( U101_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( tt ) = 1
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( x_2(x_1, x_2) ) = x_1 + 2
POL( U15_2(x_1, x_2) ) = x_1 + 1
POL( U16_1(x_1) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U34_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U35_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U36_1(x_1) ) = max{0, x_1 - 2}
POL( U41_2(x_1, x_2) ) = 2x_1 + 1
POL( U42_1(x_1) ) = 2x_1 + 2
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U72_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( s_1(x_1) ) = 0
POL( U91_2(x_1, x_2) ) = 2
POL( U92_1(x_1) ) = max{0, -2}
POL( U51_1(x_1) ) = 2x_1 + 1
POL( 0 ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(358)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U51(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( U11_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_3(x_1, ..., x_3) ) = 2
POL( U13_3(x_1, ..., x_3) ) = max{0, -2}
POL( U14_3(x_1, ..., x_3) ) = 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( tt ) = 2
POL( isNatKind_1(x_1) ) = 2x_1 + 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2
POL( plus_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( x_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U15_2(x_1, x_2) ) = 2x_1 + 2
POL( U16_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 2
POL( U31_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U35_2(x_1, x_2) ) = 2
POL( U36_1(x_1) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( U42_1(x_1) ) = 1
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U71_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 1}
POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( s_1(x_1) ) = x_1 + 2
POL( U91_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U92_1(x_1) ) = 2x_1 + 1
POL( U51_1(x_1) ) = x_1 + 2
POL( 0 ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 2}

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(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)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)


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

(360)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(362)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U51(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_3(x_1, ..., x_3) ) = max{0, -2}
POL( U13_3(x_1, ..., x_3) ) = 1
POL( U14_3(x_1, ..., x_3) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( tt ) = 1
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( x_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U15_2(x_1, x_2) ) = x_1 + 1
POL( U16_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U33_3(x_1, ..., x_3) ) = 2
POL( U34_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U35_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U36_1(x_1) ) = 0
POL( U41_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( U42_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = 2
POL( U72_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U81_3(x_1, ..., x_3) ) = x_1 + 2
POL( U82_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U84_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( s_1(x_1) ) = 2
POL( U91_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U92_1(x_1) ) = max{0, -2}
POL( U51_1(x_1) ) = x_1 + 2
POL( 0 ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(364)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U41(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( U11_3(x_1, ..., x_3) ) = 0
POL( U12_3(x_1, ..., x_3) ) = 0
POL( U13_3(x_1, ..., x_3) ) = max{0, -2}
POL( U14_3(x_1, ..., x_3) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( active_1(x_1) ) = x_1 + 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( x_2(x_1, x_2) ) = 2x_1 + 2
POL( U15_2(x_1, x_2) ) = 2x_1 + 1
POL( U16_1(x_1) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U32_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U33_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U35_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( U36_1(x_1) ) = max{0, 2x_1 - 2}
POL( U41_2(x_1, x_2) ) = 2x_1 + 2
POL( U42_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = 0
POL( U71_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U72_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U91_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U92_1(x_1) ) = 2
POL( U51_1(x_1) ) = 1
POL( 0 ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(366)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(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, x_1 - 1}
POL( U11_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U12_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U13_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U14_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U21_2(x_1, x_2) ) = 2x_2 + 2
POL( U22_2(x_1, x_2) ) = 2x_2 + 2
POL( mark_1(x_1) ) = 2x_1
POL( U102_3(x_1, ..., x_3) ) = x_1 + 2
POL( active_1(x_1) ) = x_1
POL( U101_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( tt ) = 2
POL( isNatKind_1(x_1) ) = 1
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 2x_1 + 1
POL( U104_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( plus_2(x_1, x_2) ) = 2x_1 + x_2 + 1
POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 1}
POL( U15_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( U16_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2
POL( U32_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2
POL( U33_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2
POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U35_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U36_1(x_1) ) = 0
POL( U41_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U42_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U82_3(x_1, ..., x_3) ) = x_1 + x_2
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 1
POL( U84_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( U91_2(x_1, x_2) ) = x_1 + 2
POL( U92_1(x_1) ) = max{0, x_1 - 2}
POL( U51_1(x_1) ) = max{0, x_1 - 2}
POL( 0 ) = 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:

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)


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

(368)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(370)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(X1, X2, X3)) -> ACTIVE(U11(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( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U12_3(x_1, ..., x_3) ) = max{0, -2}
POL( U13_3(x_1, ..., x_3) ) = max{0, -2}
POL( U14_3(x_1, ..., x_3) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = x_2 + 2
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( active_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2
POL( isNat_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( plus_2(x_1, x_2) ) = 2
POL( x_2(x_1, x_2) ) = 2
POL( U15_2(x_1, x_2) ) = 2x_1 + 1
POL( U16_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 1}
POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U34_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U35_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( U36_1(x_1) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U42_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_2(x_1, x_2) ) = x_1
POL( U72_2(x_1, x_2) ) = x_1 + 2
POL( U81_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( s_1(x_1) ) = 1
POL( U91_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U92_1(x_1) ) = 1
POL( U51_1(x_1) ) = max{0, -2}
POL( 0 ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(372)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   MARK(U15(X1, X2)) -> MARK(X1)
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2))
   ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2))
   ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2))
   ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
MARK(x1)  =  x1

U12(x1, x2, x3)  =  U12

ACTIVE(x1)  =  x1

U11(x1, x2, x3)  =  U11

U13(x1, x2, x3)  =  U13

U14(x1, x2, x3)  =  U14

U15(x1, x2)  =  x1

isNat(x1)  =  isNat

U21(x1, x2)  =  U21

U22(x1, x2)  =  U22

U23(x1)  =  x1

mark(x1)  =  x1

active(x1)  =  x1


Knuth-Bendix order [KBO] with precedence:trivial

and weight map:

   isNat=1
   U12=4
   U22=2
   U14=2
   U13=3
   U21=3
   U11=5

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

   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(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)


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

(374)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3))
   MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3))
   MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3))
   MARK(U15(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

(375) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
----------------------------------------

(376)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U23(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)

The TRS R consists of the following rules:

   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N))
   active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
   mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3))
   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
   mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3))
   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
   mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3))
   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
   mark(U15(X1, X2)) -> active(U15(mark(X1), X2))
   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
   mark(U16(X)) -> active(U16(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2))
   mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3))
   active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2))
   mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3))
   active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2))
   mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3))
   active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2))
   mark(U35(X1, X2)) -> active(U35(mark(X1), X2))
   active(U35(tt, V2)) -> mark(U36(isNat(V2)))
   mark(U36(X)) -> active(U36(mark(X)))
   active(U41(tt, V2)) -> mark(U42(isNatKind(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, N)) -> mark(U72(isNatKind(N), N))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, N)) -> mark(N)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(U84(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, N)) -> mark(U92(isNatKind(N)))
   mark(U92(X)) -> active(U92(mark(X)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
   mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2))
   mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3))
   active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1)))
   mark(U51(X)) -> active(U51(mark(X)))
   active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(plus(N, 0)) -> mark(U71(isNat(N), N))
   mark(U71(X1, X2)) -> active(U71(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U81(isNat(M), M, N))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U91(isNat(N), N))
   mark(U91(X1, X2)) -> active(U91(mark(X1), X2))
   active(x(N, s(M))) -> mark(U101(isNat(M), M, N))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U16(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U36(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U51(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U92(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   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)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U12(X1, mark(X2), X3) -> U12(X1, X2, X3)
   U12(mark(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, X2, mark(X3)) -> U12(X1, X2, X3)
   U12(active(X1), X2, X3) -> U12(X1, X2, X3)
   U12(X1, active(X2), X3) -> U12(X1, X2, X3)
   U12(X1, X2, active(X3)) -> U12(X1, X2, X3)
   U13(X1, mark(X2), X3) -> U13(X1, X2, X3)
   U13(mark(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, X2, mark(X3)) -> U13(X1, X2, X3)
   U13(active(X1), X2, X3) -> U13(X1, X2, X3)
   U13(X1, active(X2), X3) -> U13(X1, X2, X3)
   U13(X1, X2, active(X3)) -> U13(X1, X2, X3)
   U14(X1, mark(X2), X3) -> U14(X1, X2, X3)
   U14(mark(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, X2, mark(X3)) -> U14(X1, X2, X3)
   U14(active(X1), X2, X3) -> U14(X1, X2, X3)
   U14(X1, active(X2), X3) -> U14(X1, X2, X3)
   U14(X1, X2, active(X3)) -> U14(X1, X2, X3)
   U15(X1, mark(X2)) -> U15(X1, X2)
   U15(mark(X1), X2) -> U15(X1, X2)
   U15(active(X1), X2) -> U15(X1, X2)
   U15(X1, active(X2)) -> U15(X1, X2)
   U16(active(X)) -> U16(X)
   U16(mark(X)) -> U16(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), X3) -> U32(X1, X2, X3)
   U32(mark(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, X2, mark(X3)) -> U32(X1, X2, X3)
   U32(active(X1), X2, X3) -> U32(X1, X2, X3)
   U32(X1, active(X2), X3) -> U32(X1, X2, X3)
   U32(X1, X2, active(X3)) -> U32(X1, X2, X3)
   U33(X1, mark(X2), X3) -> U33(X1, X2, X3)
   U33(mark(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, X2, mark(X3)) -> U33(X1, X2, X3)
   U33(active(X1), X2, X3) -> U33(X1, X2, X3)
   U33(X1, active(X2), X3) -> U33(X1, X2, X3)
   U33(X1, X2, active(X3)) -> U33(X1, X2, X3)
   U34(X1, mark(X2), X3) -> U34(X1, X2, X3)
   U34(mark(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, X2, mark(X3)) -> U34(X1, X2, X3)
   U34(active(X1), X2, X3) -> U34(X1, X2, X3)
   U34(X1, active(X2), X3) -> U34(X1, X2, X3)
   U34(X1, X2, active(X3)) -> U34(X1, X2, X3)
   U35(X1, mark(X2)) -> U35(X1, X2)
   U35(mark(X1), X2) -> U35(X1, X2)
   U35(active(X1), X2) -> U35(X1, X2)
   U35(X1, active(X2)) -> U35(X1, X2)
   U36(active(X)) -> U36(X)
   U36(mark(X)) -> U36(X)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U82(X1, mark(X2), X3) -> U82(X1, X2, X3)
   U82(mark(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, X2, mark(X3)) -> U82(X1, X2, X3)
   U82(active(X1), X2, X3) -> U82(X1, X2, X3)
   U82(X1, active(X2), X3) -> U82(X1, X2, X3)
   U82(X1, X2, active(X3)) -> U82(X1, X2, X3)
   U83(X1, mark(X2), X3) -> U83(X1, X2, X3)
   U83(mark(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, X2, mark(X3)) -> U83(X1, X2, X3)
   U83(active(X1), X2, X3) -> U83(X1, X2, X3)
   U83(X1, active(X2), X3) -> U83(X1, X2, X3)
   U83(X1, X2, active(X3)) -> U83(X1, X2, X3)
   U84(X1, mark(X2), X3) -> U84(X1, X2, X3)
   U84(mark(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, X2, mark(X3)) -> U84(X1, X2, X3)
   U84(active(X1), X2, X3) -> U84(X1, X2, X3)
   U84(X1, active(X2), X3) -> U84(X1, X2, X3)
   U84(X1, X2, active(X3)) -> U84(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(active(X)) -> U92(X)
   U92(mark(X)) -> U92(X)
   U11(X1, mark(X2), X3) -> U11(X1, X2, X3)
   U11(mark(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, X2, mark(X3)) -> U11(X1, X2, X3)
   U11(active(X1), X2, X3) -> U11(X1, X2, X3)
   U11(X1, active(X2), X3) -> U11(X1, X2, X3)
   U11(X1, X2, active(X3)) -> U11(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)
   U31(X1, mark(X2), X3) -> U31(X1, X2, X3)
   U31(mark(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, X2, mark(X3)) -> U31(X1, X2, X3)
   U31(active(X1), X2, X3) -> U31(X1, X2, X3)
   U31(X1, active(X2), X3) -> U31(X1, X2, X3)
   U31(X1, X2, active(X3)) -> U31(X1, X2, X3)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(active(X)) -> U51(X)
   U51(mark(X)) -> U51(X)
   U71(X1, mark(X2)) -> U71(X1, X2)
   U71(mark(X1), X2) -> U71(X1, X2)
   U71(active(X1), X2) -> U71(X1, X2)
   U71(X1, active(X2)) -> U71(X1, X2)
   U81(X1, mark(X2), X3) -> U81(X1, X2, X3)
   U81(mark(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, X2, mark(X3)) -> U81(X1, X2, X3)
   U81(active(X1), X2, X3) -> U81(X1, X2, X3)
   U81(X1, active(X2), X3) -> U81(X1, X2, X3)
   U81(X1, X2, active(X3)) -> U81(X1, X2, X3)
   U91(X1, mark(X2)) -> U91(X1, X2)
   U91(mark(X1), X2) -> U91(X1, X2)
   U91(active(X1), X2) -> U91(X1, X2)
   U91(X1, active(X2)) -> U91(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

(378)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U23(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))

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

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

   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U11(tt, x0, x1))
   active(U12(tt, x0, x1))
   active(U13(tt, x0, x1))
   active(U14(tt, x0, x1))
   active(U15(tt, x0))
   active(U16(tt))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0, x1))
   active(U32(tt, x0, x1))
   active(U33(tt, x0, x1))
   active(U34(tt, x0, x1))
   active(U35(tt, x0))
   active(U36(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0))
   active(U72(tt, x0))
   active(U81(tt, x0, x1))
   active(U82(tt, x0, x1))
   active(U83(tt, x0, x1))
   active(U84(tt, x0, x1))
   active(U91(tt, x0))
   active(U92(tt))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(plus(x0, x1)))
   active(isNatKind(s(x0)))
   active(isNatKind(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNat(x0))
   mark(U104(x0, x1, x2))
   mark(plus(x0, x1))
   mark(x(x0, x1))
   mark(U11(x0, x1, x2))
   mark(U12(x0, x1, x2))
   mark(U13(x0, x1, x2))
   mark(U14(x0, x1, x2))
   mark(U15(x0, x1))
   mark(U16(x0))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1, x2))
   mark(U32(x0, x1, x2))
   mark(U33(x0, x1, x2))
   mark(U34(x0, x1, x2))
   mark(U35(x0, x1))
   mark(U36(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(U51(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1))
   mark(U72(x0, x1))
   mark(U81(x0, x1, x2))
   mark(U82(x0, x1, x2))
   mark(U83(x0, x1, x2))
   mark(U84(x0, x1, x2))
   mark(s(x0))
   mark(U91(x0, x1))
   mark(U92(x0))
   mark(0)
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNat(mark(x0))
   isNat(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))
   U11(mark(x0), x1, x2)
   U11(x0, mark(x1), x2)
   U11(x0, x1, mark(x2))
   U11(active(x0), x1, x2)
   U11(x0, active(x1), x2)
   U11(x0, x1, active(x2))
   U12(mark(x0), x1, x2)
   U12(x0, mark(x1), x2)
   U12(x0, x1, mark(x2))
   U12(active(x0), x1, x2)
   U12(x0, active(x1), x2)
   U12(x0, x1, active(x2))
   U13(mark(x0), x1, x2)
   U13(x0, mark(x1), x2)
   U13(x0, x1, mark(x2))
   U13(active(x0), x1, x2)
   U13(x0, active(x1), x2)
   U13(x0, x1, active(x2))
   U14(mark(x0), x1, x2)
   U14(x0, mark(x1), x2)
   U14(x0, x1, mark(x2))
   U14(active(x0), x1, x2)
   U14(x0, active(x1), x2)
   U14(x0, x1, active(x2))
   U16(mark(x0))
   U16(active(x0))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U31(mark(x0), x1, x2)
   U31(x0, mark(x1), x2)
   U31(x0, x1, mark(x2))
   U31(active(x0), x1, x2)
   U31(x0, active(x1), x2)
   U31(x0, x1, active(x2))
   U32(mark(x0), x1, x2)
   U32(x0, mark(x1), x2)
   U32(x0, x1, mark(x2))
   U32(active(x0), x1, x2)
   U32(x0, active(x1), x2)
   U32(x0, x1, active(x2))
   U33(mark(x0), x1, x2)
   U33(x0, mark(x1), x2)
   U33(x0, x1, mark(x2))
   U33(active(x0), x1, x2)
   U33(x0, active(x1), x2)
   U33(x0, x1, active(x2))
   U34(mark(x0), x1, x2)
   U34(x0, mark(x1), x2)
   U34(x0, x1, mark(x2))
   U34(active(x0), x1, x2)
   U34(x0, active(x1), x2)
   U34(x0, x1, active(x2))
   U35(mark(x0), x1)
   U35(x0, mark(x1))
   U35(active(x0), x1)
   U35(x0, active(x1))
   U36(mark(x0))
   U36(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   U51(mark(x0))
   U51(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1)
   U71(x0, mark(x1))
   U71(active(x0), x1)
   U71(x0, active(x1))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   U81(mark(x0), x1, x2)
   U81(x0, mark(x1), x2)
   U81(x0, x1, mark(x2))
   U81(active(x0), x1, x2)
   U81(x0, active(x1), x2)
   U81(x0, x1, active(x2))
   U82(mark(x0), x1, x2)
   U82(x0, mark(x1), x2)
   U82(x0, x1, mark(x2))
   U82(active(x0), x1, x2)
   U82(x0, active(x1), x2)
   U82(x0, x1, active(x2))
   U83(mark(x0), x1, x2)
   U83(x0, mark(x1), x2)
   U83(x0, x1, mark(x2))
   U83(active(x0), x1, x2)
   U83(x0, active(x1), x2)
   U83(x0, x1, active(x2))
   U84(mark(x0), x1, x2)
   U84(x0, mark(x1), x2)
   U84(x0, x1, mark(x2))
   U84(active(x0), x1, x2)
   U84(x0, active(x1), x2)
   U84(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   U91(mark(x0), x1)
   U91(x0, mark(x1))
   U91(active(x0), x1)
   U91(x0, active(x1))
   U92(mark(x0))
   U92(active(x0))


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

(380)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U23(X)) -> MARK(X)
   MARK(U15(X1, X2)) -> MARK(X1)

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

   U15(mark(x0), x1)
   U15(x0, mark(x1))
   U15(active(x0), x1)
   U15(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))

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

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

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


*MARK(U15(X1, X2)) -> MARK(X1)
The graph contains the following edges 1 > 1


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

(382)
YES