/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 350 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) UsableRulesProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) UsableRulesProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES
    (25) QDP
        (26) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (29) YES
    (30) QDP
        (31) UsableRulesProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (34) YES
    (35) QDP
        (36) UsableRulesProof [EQUIVALENT, 1 ms]
        (37) QDP
        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (39) YES
    (40) QDP
        (41) UsableRulesProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (44) YES
    (45) QDP
        (46) UsableRulesProof [EQUIVALENT, 0 ms]
        (47) QDP
        (48) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (49) YES
    (50) QDP
        (51) UsableRulesProof [EQUIVALENT, 0 ms]
        (52) QDP
        (53) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (54) YES
    (55) QDP
        (56) UsableRulesProof [EQUIVALENT, 0 ms]
        (57) QDP
        (58) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (59) YES
    (60) QDP
        (61) UsableRulesProof [EQUIVALENT, 0 ms]
        (62) QDP
        (63) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (64) YES
    (65) QDP
        (66) UsableRulesProof [EQUIVALENT, 0 ms]
        (67) QDP
        (68) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (69) YES
    (70) QDP
        (71) UsableRulesProof [EQUIVALENT, 0 ms]
        (72) QDP
        (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (74) YES
    (75) QDP
        (76) UsableRulesProof [EQUIVALENT, 0 ms]
        (77) QDP
        (78) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (79) YES
    (80) QDP
        (81) UsableRulesProof [EQUIVALENT, 0 ms]
        (82) QDP
        (83) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (84) YES
    (85) QDP
        (86) UsableRulesProof [EQUIVALENT, 0 ms]
        (87) QDP
        (88) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (89) YES
    (90) QDP
        (91) UsableRulesProof [EQUIVALENT, 0 ms]
        (92) QDP
        (93) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (94) YES
    (95) QDP
        (96) UsableRulesProof [EQUIVALENT, 0 ms]
        (97) QDP
        (98) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (99) YES
    (100) QDP
        (101) UsableRulesProof [EQUIVALENT, 0 ms]
        (102) QDP
        (103) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (104) YES
    (105) QDP
        (106) UsableRulesProof [EQUIVALENT, 0 ms]
        (107) QDP
        (108) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (109) YES
    (110) QDP
        (111) UsableRulesProof [EQUIVALENT, 0 ms]
        (112) QDP
        (113) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (114) YES
    (115) QDP
        (116) UsableRulesProof [EQUIVALENT, 0 ms]
        (117) QDP
        (118) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (119) YES
    (120) QDP
        (121) UsableRulesProof [EQUIVALENT, 0 ms]
        (122) QDP
        (123) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (124) YES
    (125) QDP
        (126) UsableRulesProof [EQUIVALENT, 0 ms]
        (127) QDP
        (128) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (129) YES
    (130) QDP
        (131) UsableRulesProof [EQUIVALENT, 0 ms]
        (132) QDP
        (133) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (134) YES
    (135) QDP
        (136) UsableRulesProof [EQUIVALENT, 0 ms]
        (137) QDP
        (138) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (139) YES
    (140) QDP
        (141) UsableRulesProof [EQUIVALENT, 0 ms]
        (142) QDP
        (143) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (144) YES
    (145) QDP
        (146) UsableRulesProof [EQUIVALENT, 0 ms]
        (147) QDP
        (148) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (149) YES
    (150) QDP
        (151) UsableRulesProof [EQUIVALENT, 0 ms]
        (152) QDP
        (153) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (154) YES
    (155) QDP
        (156) UsableRulesProof [EQUIVALENT, 0 ms]
        (157) QDP
        (158) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (159) YES
    (160) QDP
        (161) UsableRulesProof [EQUIVALENT, 0 ms]
        (162) QDP
        (163) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (164) YES
    (165) QDP
        (166) UsableRulesProof [EQUIVALENT, 0 ms]
        (167) QDP
        (168) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (169) YES
    (170) QDP
        (171) UsableRulesProof [EQUIVALENT, 0 ms]
        (172) QDP
        (173) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (174) YES
    (175) QDP
        (176) UsableRulesProof [EQUIVALENT, 0 ms]
        (177) QDP
        (178) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (179) YES
    (180) QDP
        (181) UsableRulesProof [EQUIVALENT, 0 ms]
        (182) QDP
        (183) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (184) YES
    (185) QDP
        (186) UsableRulesProof [EQUIVALENT, 0 ms]
        (187) QDP
        (188) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (189) YES
    (190) QDP
        (191) QDPOrderProof [EQUIVALENT, 2036 ms]
        (192) QDP
        (193) QDPOrderProof [EQUIVALENT, 1471 ms]
        (194) QDP
        (195) QDPOrderProof [EQUIVALENT, 2672 ms]
        (196) QDP
        (197) QDPOrderProof [EQUIVALENT, 1535 ms]
        (198) QDP
        (199) DependencyGraphProof [EQUIVALENT, 0 ms]
        (200) QDP
        (201) QDPOrderProof [EQUIVALENT, 914 ms]
        (202) QDP
        (203) QDPOrderProof [EQUIVALENT, 1832 ms]
        (204) QDP
        (205) QDPOrderProof [EQUIVALENT, 1301 ms]
        (206) QDP
        (207) QDPOrderProof [EQUIVALENT, 2328 ms]
        (208) QDP
        (209) QDPOrderProof [EQUIVALENT, 1070 ms]
        (210) QDP
        (211) QDPOrderProof [EQUIVALENT, 1230 ms]
        (212) QDP
        (213) QDPOrderProof [EQUIVALENT, 2062 ms]
        (214) QDP
        (215) QDPOrderProof [EQUIVALENT, 2083 ms]
        (216) QDP
        (217) QDPOrderProof [EQUIVALENT, 2463 ms]
        (218) QDP
        (219) QDPOrderProof [EQUIVALENT, 1069 ms]
        (220) QDP
        (221) QDPOrderProof [EQUIVALENT, 2705 ms]
        (222) QDP
        (223) QDPOrderProof [EQUIVALENT, 1235 ms]
        (224) QDP
        (225) QDPOrderProof [EQUIVALENT, 1774 ms]
        (226) QDP
        (227) QDPOrderProof [EQUIVALENT, 1079 ms]
        (228) QDP
        (229) QDPOrderProof [EQUIVALENT, 1360 ms]
        (230) QDP
        (231) QDPOrderProof [EQUIVALENT, 2309 ms]
        (232) QDP
        (233) QDPOrderProof [EQUIVALENT, 948 ms]
        (234) QDP
        (235) QDPOrderProof [EQUIVALENT, 1111 ms]
        (236) QDP
        (237) QDPOrderProof [EQUIVALENT, 902 ms]
        (238) QDP
        (239) QDPOrderProof [EQUIVALENT, 1085 ms]
        (240) QDP
        (241) QDPOrderProof [EQUIVALENT, 2228 ms]
        (242) QDP
        (243) QDPOrderProof [EQUIVALENT, 997 ms]
        (244) QDP
        (245) QDPOrderProof [EQUIVALENT, 1117 ms]
        (246) QDP
        (247) QDPOrderProof [EQUIVALENT, 1005 ms]
        (248) QDP
        (249) QDPOrderProof [EQUIVALENT, 994 ms]
        (250) QDP
        (251) QDPOrderProof [EQUIVALENT, 1012 ms]
        (252) QDP
        (253) QDPOrderProof [EQUIVALENT, 1171 ms]
        (254) QDP
        (255) QDPOrderProof [EQUIVALENT, 949 ms]
        (256) QDP
        (257) QDPOrderProof [EQUIVALENT, 1199 ms]
        (258) QDP
        (259) QDPOrderProof [EQUIVALENT, 826 ms]
        (260) QDP
        (261) QDPOrderProof [EQUIVALENT, 1115 ms]
        (262) QDP
        (263) QDPOrderProof [EQUIVALENT, 875 ms]
        (264) QDP
        (265) QDPOrderProof [EQUIVALENT, 1253 ms]
        (266) QDP
        (267) QDPOrderProof [EQUIVALENT, 1751 ms]
        (268) QDP
        (269) QDPOrderProof [EQUIVALENT, 997 ms]
        (270) QDP
        (271) QDPOrderProof [EQUIVALENT, 1101 ms]
        (272) QDP
        (273) QDPOrderProof [EQUIVALENT, 799 ms]
        (274) QDP
        (275) QDPOrderProof [EQUIVALENT, 908 ms]
        (276) QDP
        (277) QDPOrderProof [EQUIVALENT, 805 ms]
        (278) QDP
        (279) QDPOrderProof [EQUIVALENT, 911 ms]
        (280) QDP
        (281) QDPOrderProof [EQUIVALENT, 703 ms]
        (282) QDP
        (283) QDPOrderProof [EQUIVALENT, 776 ms]
        (284) QDP
        (285) QDPOrderProof [EQUIVALENT, 790 ms]
        (286) QDP
        (287) QDPOrderProof [EQUIVALENT, 819 ms]
        (288) QDP
        (289) QDPOrderProof [EQUIVALENT, 929 ms]
        (290) QDP
        (291) QDPOrderProof [EQUIVALENT, 701 ms]
        (292) QDP
        (293) QDPOrderProof [EQUIVALENT, 2345 ms]
        (294) QDP
        (295) QDPOrderProof [EQUIVALENT, 835 ms]
        (296) QDP
        (297) QDPOrderProof [EQUIVALENT, 762 ms]
        (298) QDP
        (299) QDPOrderProof [EQUIVALENT, 732 ms]
        (300) QDP
        (301) QDPOrderProof [EQUIVALENT, 799 ms]
        (302) QDP
        (303) QDPOrderProof [EQUIVALENT, 613 ms]
        (304) QDP
        (305) QDPOrderProof [EQUIVALENT, 682 ms]
        (306) QDP
        (307) QDPOrderProof [EQUIVALENT, 360 ms]
        (308) QDP
        (309) DependencyGraphProof [EQUIVALENT, 0 ms]
        (310) TRUE


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

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

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.

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

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

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

   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(zeros) -> CONS(0, zeros)
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U11(tt, V1)) -> U12^1(isNatIListKind(V1), V1)
   ACTIVE(U11(tt, V1)) -> ISNATILISTKIND(V1)
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U12(tt, V1)) -> U13^1(isNatList(V1))
   ACTIVE(U12(tt, V1)) -> ISNATLIST(V1)
   ACTIVE(U13(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, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U31(tt, V)) -> U32^1(isNatIListKind(V), V)
   ACTIVE(U31(tt, V)) -> ISNATILISTKIND(V)
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U32(tt, V)) -> U33^1(isNatList(V))
   ACTIVE(U32(tt, V)) -> ISNATLIST(V)
   ACTIVE(U33(tt)) -> MARK(tt)
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U41(tt, V1, V2)) -> U42^1(isNatKind(V1), V1, V2)
   ACTIVE(U41(tt, V1, V2)) -> ISNATKIND(V1)
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> U43^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U42(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> U44^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U43(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U44(tt, V1, V2)) -> U45^1(isNat(V1), V2)
   ACTIVE(U44(tt, V1, V2)) -> ISNAT(V1)
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U45(tt, V2)) -> U46^1(isNatIList(V2))
   ACTIVE(U45(tt, V2)) -> ISNATILIST(V2)
   ACTIVE(U46(tt)) -> MARK(tt)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U51(tt, V2)) -> U52^1(isNatIListKind(V2))
   ACTIVE(U51(tt, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U52(tt)) -> MARK(tt)
   ACTIVE(U61(tt)) -> MARK(tt)
   ACTIVE(U71(tt)) -> MARK(tt)
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   ACTIVE(U81(tt, V1, V2)) -> U82^1(isNatKind(V1), V1, V2)
   ACTIVE(U81(tt, V1, V2)) -> ISNATKIND(V1)
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   ACTIVE(U82(tt, V1, V2)) -> U83^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U82(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U83(tt, V1, V2)) -> U84^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U83(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U84(tt, V1, V2)) -> U85^1(isNat(V1), V2)
   ACTIVE(U84(tt, V1, V2)) -> ISNAT(V1)
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U85(tt, V2)) -> U86^1(isNatList(V2))
   ACTIVE(U85(tt, V2)) -> ISNATLIST(V2)
   ACTIVE(U86(tt)) -> MARK(tt)
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U91(tt, L, N)) -> U92^1(isNatIListKind(L), L, N)
   ACTIVE(U91(tt, L, N)) -> ISNATILISTKIND(L)
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U92(tt, L, N)) -> U93^1(isNat(N), L, N)
   ACTIVE(U92(tt, L, N)) -> ISNAT(N)
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   ACTIVE(U93(tt, L, N)) -> U94^1(isNatKind(N), L)
   ACTIVE(U93(tt, L, N)) -> ISNATKIND(N)
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U94(tt, L)) -> S(length(L))
   ACTIVE(U94(tt, L)) -> LENGTH(L)
   ACTIVE(isNat(0)) -> MARK(tt)
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   ACTIVE(isNat(length(V1))) -> U11^1(isNatIListKind(V1), V1)
   ACTIVE(isNat(length(V1))) -> ISNATILISTKIND(V1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(s(V1))) -> U21^1(isNatKind(V1), V1)
   ACTIVE(isNat(s(V1))) -> ISNATKIND(V1)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   ACTIVE(isNatIList(V)) -> U31^1(isNatIListKind(V), V)
   ACTIVE(isNatIList(V)) -> ISNATILISTKIND(V)
   ACTIVE(isNatIList(zeros)) -> MARK(tt)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> U41^1(isNatKind(V1), V1, V2)
   ACTIVE(isNatIList(cons(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatIListKind(nil)) -> MARK(tt)
   ACTIVE(isNatIListKind(zeros)) -> MARK(tt)
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> U51^1(isNatKind(V1), V2)
   ACTIVE(isNatIListKind(cons(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatKind(0)) -> MARK(tt)
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(length(V1))) -> U61^1(isNatIListKind(V1))
   ACTIVE(isNatKind(length(V1))) -> ISNATILISTKIND(V1)
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> U71^1(isNatKind(V1))
   ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1)
   ACTIVE(isNatList(nil)) -> MARK(tt)
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(isNatList(cons(V1, V2))) -> U81^1(isNatKind(V1), V1, V2)
   ACTIVE(isNatList(cons(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(length(nil)) -> MARK(0)
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))
   ACTIVE(length(cons(N, L))) -> U91^1(isNatList(L), L, N)
   ACTIVE(length(cons(N, L))) -> ISNATLIST(L)
   MARK(zeros) -> ACTIVE(zeros)
   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(cons(X1, X2)) -> CONS(mark(X1), X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(0) -> ACTIVE(0)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> U11^1(mark(X1), X2)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(tt) -> ACTIVE(tt)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   MARK(U12(X1, X2)) -> U12^1(mark(X1), X2)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   MARK(U13(X)) -> U13^1(mark(X))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(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(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   MARK(U23(X)) -> U23^1(mark(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> U31^1(mark(X1), X2)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   MARK(U32(X1, X2)) -> U32^1(mark(X1), X2)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   MARK(U33(X)) -> U33^1(mark(X))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> U41^1(mark(X1), X2, X3)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   MARK(U42(X1, X2, X3)) -> U42^1(mark(X1), X2, X3)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U43(X1, X2, X3)) -> U43^1(mark(X1), X2, X3)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> U44^1(mark(X1), X2, X3)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U45(X1, X2)) -> U45^1(mark(X1), X2)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   MARK(U46(X)) -> U46^1(mark(X))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> U51^1(mark(X1), X2)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   MARK(U52(X)) -> U52^1(mark(X))
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> ACTIVE(U61(mark(X)))
   MARK(U61(X)) -> U61^1(mark(X))
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   MARK(U71(X)) -> U71^1(mark(X))
   MARK(U71(X)) -> MARK(X)
   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(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   MARK(U85(X1, X2)) -> U85^1(mark(X1), X2)
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> ACTIVE(U86(mark(X)))
   MARK(U86(X)) -> U86^1(mark(X))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> U91^1(mark(X1), X2, X3)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> U92^1(mark(X1), X2, X3)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> U93^1(mark(X1), X2, X3)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(U94(X1, X2)) -> U94^1(mark(X1), X2)
   MARK(U94(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(s(X)) -> S(mark(X))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> LENGTH(mark(X))
   MARK(length(X)) -> MARK(X)
   MARK(nil) -> ACTIVE(nil)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)
   U12^1(mark(X1), X2) -> U12^1(X1, X2)
   U12^1(X1, mark(X2)) -> U12^1(X1, X2)
   U12^1(active(X1), X2) -> U12^1(X1, X2)
   U12^1(X1, active(X2)) -> U12^1(X1, X2)
   ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)
   ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
   U13^1(mark(X)) -> U13^1(X)
   U13^1(active(X)) -> U13^1(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)
   ISNATLIST(active(X)) -> ISNATLIST(X)
   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)
   ISNATKIND(mark(X)) -> ISNATKIND(X)
   ISNATKIND(active(X)) -> ISNATKIND(X)
   U23^1(mark(X)) -> U23^1(X)
   U23^1(active(X)) -> U23^1(X)
   ISNAT(mark(X)) -> ISNAT(X)
   ISNAT(active(X)) -> ISNAT(X)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)
   U32^1(mark(X1), X2) -> U32^1(X1, X2)
   U32^1(X1, mark(X2)) -> U32^1(X1, X2)
   U32^1(active(X1), X2) -> U32^1(X1, X2)
   U32^1(X1, active(X2)) -> U32^1(X1, X2)
   U33^1(mark(X)) -> U33^1(X)
   U33^1(active(X)) -> U33^1(X)
   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
   U43^1(mark(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, mark(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, mark(X3)) -> U43^1(X1, X2, X3)
   U43^1(active(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, active(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, active(X3)) -> U43^1(X1, X2, X3)
   U44^1(mark(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, mark(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, mark(X3)) -> U44^1(X1, X2, X3)
   U44^1(active(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, active(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, active(X3)) -> U44^1(X1, X2, X3)
   U45^1(mark(X1), X2) -> U45^1(X1, X2)
   U45^1(X1, mark(X2)) -> U45^1(X1, X2)
   U45^1(active(X1), X2) -> U45^1(X1, X2)
   U45^1(X1, active(X2)) -> U45^1(X1, X2)
   U46^1(mark(X)) -> U46^1(X)
   U46^1(active(X)) -> U46^1(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)
   ISNATILIST(active(X)) -> ISNATILIST(X)
   U51^1(mark(X1), X2) -> U51^1(X1, X2)
   U51^1(X1, mark(X2)) -> U51^1(X1, X2)
   U51^1(active(X1), X2) -> U51^1(X1, X2)
   U51^1(X1, active(X2)) -> U51^1(X1, X2)
   U52^1(mark(X)) -> U52^1(X)
   U52^1(active(X)) -> U52^1(X)
   U61^1(mark(X)) -> U61^1(X)
   U61^1(active(X)) -> U61^1(X)
   U71^1(mark(X)) -> U71^1(X)
   U71^1(active(X)) -> U71^1(X)
   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)
   U85^1(mark(X1), X2) -> U85^1(X1, X2)
   U85^1(X1, mark(X2)) -> U85^1(X1, X2)
   U85^1(active(X1), X2) -> U85^1(X1, X2)
   U85^1(X1, active(X2)) -> U85^1(X1, X2)
   U86^1(mark(X)) -> U86^1(X)
   U86^1(active(X)) -> U86^1(X)
   U91^1(mark(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, mark(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, mark(X3)) -> U91^1(X1, X2, X3)
   U91^1(active(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, active(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, active(X3)) -> U91^1(X1, X2, X3)
   U92^1(mark(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, mark(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, mark(X3)) -> U92^1(X1, X2, X3)
   U92^1(active(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, active(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, active(X3)) -> U92^1(X1, X2, X3)
   U93^1(mark(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, mark(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, mark(X3)) -> U93^1(X1, X2, X3)
   U93^1(active(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, active(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, active(X3)) -> U93^1(X1, X2, X3)
   U94^1(mark(X1), X2) -> U94^1(X1, X2)
   U94^1(X1, mark(X2)) -> U94^1(X1, X2)
   U94^1(active(X1), X2) -> U94^1(X1, X2)
   U94^1(X1, active(X2)) -> U94^1(X1, X2)
   S(mark(X)) -> S(X)
   S(active(X)) -> S(X)
   LENGTH(mark(X)) -> LENGTH(X)
   LENGTH(active(X)) -> LENGTH(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 38 SCCs with 111 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

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

   LENGTH(active(X)) -> LENGTH(X)
   LENGTH(mark(X)) -> LENGTH(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

   LENGTH(active(X)) -> LENGTH(X)
   LENGTH(mark(X)) -> LENGTH(X)

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

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

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


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


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

(9)
YES

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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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

(19)
YES

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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


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


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

(24)
YES

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

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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


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


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

(29)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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


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


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

(34)
YES

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

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

   U86^1(active(X)) -> U86^1(X)
   U86^1(mark(X)) -> U86^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

   U86^1(active(X)) -> U86^1(X)
   U86^1(mark(X)) -> U86^1(X)

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

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

From the DPs we obtained the following set of size-change graphs:
*U86^1(active(X)) -> U86^1(X)
The graph contains the following edges 1 > 1


*U86^1(mark(X)) -> U86^1(X)
The graph contains the following edges 1 > 1


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

(39)
YES

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

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

   U85^1(X1, mark(X2)) -> U85^1(X1, X2)
   U85^1(mark(X1), X2) -> U85^1(X1, X2)
   U85^1(active(X1), X2) -> U85^1(X1, X2)
   U85^1(X1, active(X2)) -> U85^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

   U85^1(X1, mark(X2)) -> U85^1(X1, X2)
   U85^1(mark(X1), X2) -> U85^1(X1, X2)
   U85^1(active(X1), X2) -> U85^1(X1, X2)
   U85^1(X1, active(X2)) -> U85^1(X1, X2)

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

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

From the DPs we obtained the following set of size-change graphs:
*U85^1(X1, mark(X2)) -> U85^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U85^1(mark(X1), X2) -> U85^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U85^1(active(X1), X2) -> U85^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U85^1(X1, active(X2)) -> U85^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(44)
YES

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

(45)
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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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


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

(49)
YES

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

(50)
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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(52)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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


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

(54)
YES

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

(55)
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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(57)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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


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

(59)
YES

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

(60)
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(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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


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

(64)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(67)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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

(69)
YES

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

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

   U61^1(active(X)) -> U61^1(X)
   U61^1(mark(X)) -> U61^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

   U61^1(active(X)) -> U61^1(X)
   U61^1(mark(X)) -> U61^1(X)

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

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

From the DPs we obtained the following set of size-change graphs:
*U61^1(active(X)) -> U61^1(X)
The graph contains the following edges 1 > 1


*U61^1(mark(X)) -> U61^1(X)
The graph contains the following edges 1 > 1


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

(74)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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


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

(79)
YES

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

(80)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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


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


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


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

(84)
YES

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

(85)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(active(X)) -> ISNATILIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(87)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(active(X)) -> ISNATILIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)

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

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

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


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


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

(89)
YES

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

(90)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(92)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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

(94)
YES

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

(95)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(97)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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

(99)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(102)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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


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


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

(104)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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


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


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


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


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


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


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

(109)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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


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


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


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


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


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

(114)
YES

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

(115)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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


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


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


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


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


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

(119)
YES

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

(120)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(122)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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

(124)
YES

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

(125)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(127)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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

(129)
YES

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

(130)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(132)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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


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


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


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

(134)
YES

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

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

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(137)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

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

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

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


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

(139)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

(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:
*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


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

(144)
YES

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

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

   ISNATKIND(active(X)) -> ISNATKIND(X)
   ISNATKIND(mark(X)) -> ISNATKIND(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

   ISNATKIND(active(X)) -> ISNATKIND(X)
   ISNATKIND(mark(X)) -> ISNATKIND(X)

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

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

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


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

(149)
YES

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

(150)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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

(154)
YES

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

(155)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(157)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

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


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

(159)
YES

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

(160)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(162)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)

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

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

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


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


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

(164)
YES

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

(165)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U13^1(active(X)) -> U13^1(X)
   U13^1(mark(X)) -> U13^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(167)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U13^1(active(X)) -> U13^1(X)
   U13^1(mark(X)) -> U13^1(X)

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

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

From the DPs we obtained the following set of size-change graphs:
*U13^1(active(X)) -> U13^1(X)
The graph contains the following edges 1 > 1


*U13^1(mark(X)) -> U13^1(X)
The graph contains the following edges 1 > 1


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

(169)
YES

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

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

   ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
   ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(172)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
   ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)

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

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

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


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


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

(174)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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


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


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


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


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

(179)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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


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


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


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

(184)
YES

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

(185)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

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

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

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

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

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


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


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


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


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

(189)
YES

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

(190)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> ACTIVE(U61(mark(X)))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> ACTIVE(U86(mark(X)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(U94(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   MARK(U61(X)) -> ACTIVE(U61(mark(X)))
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   MARK(U86(X)) -> ACTIVE(U86(mark(X)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U13_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = 0
POL( U61_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 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) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U86_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 1
POL( mark_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( isNatIListKind_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

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

   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   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), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(192)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U45(X1, X2)) -> MARK(X1)
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U61(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(U94(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

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

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( 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( U85_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = max{0, -2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( isNatIListKind_1(x_1) ) = 2
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_1(x_1) ) = 2x_1 + 2
POL( U71_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

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

   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   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)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   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), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

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

   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U45(X1, X2)) -> MARK(X1)
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U61(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(U94(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2)) -> MARK(X1)
   MARK(length(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) ) = 2x_1
POL( MARK_1(x_1) ) = 2x_1
POL( U11_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1
POL( U12_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( U31_2(x_1, x_2) ) = x_1 + 2x_2
POL( U32_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
POL( U52_1(x_1) ) = x_1
POL( U61_1(x_1) ) = 2x_1
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U21_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( U22_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
POL( U45_2(x_1, x_2) ) = x_1 + 2x_2
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3
POL( U82_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3
POL( U85_2(x_1, x_2) ) = x_1 + 2x_2
POL( U91_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 + 1
POL( U93_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 1
POL( U94_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1
POL( length_1(x_1) ) = 2x_1 + 1
POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( U13_1(x_1) ) = x_1
POL( isNatList_1(x_1) ) = 2x_1
POL( isNatKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 2x_1
POL( U33_1(x_1) ) = x_1
POL( U46_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 2x_1
POL( U86_1(x_1) ) = x_1
POL( s_1(x_1) ) = x_1
POL( U71_1(x_1) ) = 2x_1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(isNat(X)) -> active(isNat(X))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U94(tt, L)) -> mark(s(length(L)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U61(X)) -> active(U61(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   mark(U85(X1, X2)) -> active(U85(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   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)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U86(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)


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

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

   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U45(X1, X2)) -> MARK(X1)
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U61(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(zeros) -> ACTIVE(zeros)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   MARK(U12(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_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U12_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U84_3(x_1, ..., x_3) ) = x_1 + 1
POL( U85_2(x_1, x_2) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = x_1 + 1
POL( active_1(x_1) ) = 2x_1 + 2
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U33_1(x_1) ) = x_1 + 1
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = x_1 + 1
POL( U86_1(x_1) ) = x_1 + 1
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   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)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

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

   MARK(cons(X1, X2)) -> MARK(X1)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U45(X1, X2)) -> MARK(X1)
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U61(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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

(200)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 1
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = 2x_1 + 2
POL( U61_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 2x_1 + 2
POL( U33_1(x_1) ) = 2x_1 + 2
POL( U86_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 1
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 0
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2x_2 + 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 1
POL( U46_1(x_1) ) = 2x_1 + 2
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 0
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(202)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U13(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U32_2(x_1, x_2) ) = x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = 2x_1 + 1
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 1
POL( mark_1(x_1) ) = x_1
POL( U33_1(x_1) ) = 2x_1 + 1
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = x_1 + 1
POL( U22_2(x_1, x_2) ) = x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U81_3(x_1, ..., x_3) ) = x_1 + 1
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_2(x_1, x_2) ) = 1
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U85_2(x_1, x_2) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2x_1 + 1
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = 2x_1 + 2
POL( U13_1(x_1) ) = 2x_1 + 2
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)


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

(204)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = 2x_1 + 1
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U33_1(x_1) ) = x_1 + 1
POL( U86_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U85_2(x_1, x_2) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( zeros ) = 1
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)


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

(206)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2x_1
POL( MARK_1(x_1) ) = 2x_1
POL( U31_2(x_1, x_2) ) = x_1 + 2
POL( U32_2(x_1, x_2) ) = x_1 + 2
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U44_3(x_1, ..., x_3) ) = x_1 + 2
POL( U52_1(x_1) ) = 2x_1
POL( U61_1(x_1) ) = 2x_1
POL( U83_3(x_1, ..., x_3) ) = x_1
POL( U84_3(x_1, ..., x_3) ) = 2x_1
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U33_1(x_1) ) = 2x_1
POL( U86_1(x_1) ) = 2x_1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = x_1
POL( U22_2(x_1, x_2) ) = 2x_1
POL( U41_3(x_1, ..., x_3) ) = x_1 + 2
POL( U42_3(x_1, ..., x_3) ) = x_1 + 2
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U71_1(x_1) ) = 2x_1
POL( U81_3(x_1, ..., x_3) ) = 2x_1
POL( U82_3(x_1, ..., x_3) ) = 2x_1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = 2x_1 + 2
POL( U85_2(x_1, x_2) ) = 2x_1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = 2x_1
POL( nil ) = 2

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(isNat(X)) -> active(isNat(X))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U94(tt, L)) -> mark(s(length(L)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U61(X)) -> active(U61(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   mark(U85(X1, X2)) -> active(U85(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U86(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)


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

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

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( MARK_1(x_1) ) = 1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = 2
POL( U61_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( active_1(x_1) ) = 1
POL( mark_1(x_1) ) = 0
POL( U86_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = x_1 + 1
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U13_1(x_1) ) = x_1 + 1
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 2
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U46_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

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

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U33(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = x_1
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 0
POL( U52_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = 2x_1 + 1
POL( U83_3(x_1, ..., x_3) ) = x_1 + 1
POL( U84_3(x_1, ..., x_3) ) = x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 1
POL( mark_1(x_1) ) = x_1 + 2
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U33_1(x_1) ) = x_1 + 2
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)


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

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

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U81(X1, X2, X3)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U85(X1, X2)) -> MARK(X1)
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

(213) 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)) -> MARK(X1)
   MARK(U82(X1, X2, X3)) -> MARK(X1)
   MARK(U83(X1, X2, X3)) -> MARK(X1)
   MARK(U84(X1, X2, X3)) -> MARK(X1)
   MARK(U85(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( MARK_1(x_1) ) = x_1
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = 0
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = x_1
POL( U61_1(x_1) ) = x_1
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U86_1(x_1) ) = x_1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 1
POL( U21_2(x_1, x_2) ) = 2x_1
POL( U22_2(x_1, x_2) ) = x_1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U71_1(x_1) ) = 2x_1
POL( U81_3(x_1, ..., x_3) ) = x_1 + 1
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 2x_1
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1
POL( isNat_1(x_1) ) = 0
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(isNat(X)) -> active(isNat(X))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U94(tt, L)) -> mark(s(length(L)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U61(X)) -> active(U61(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   mark(U85(X1, X2)) -> active(U85(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U86(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)


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

(214)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( MARK_1(x_1) ) = x_1
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = x_1
POL( U52_1(x_1) ) = 2x_1
POL( U61_1(x_1) ) = x_1
POL( U83_3(x_1, ..., x_3) ) = 2x_1
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U86_1(x_1) ) = 2x_1
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 2x_1 + 2
POL( U22_2(x_1, x_2) ) = 2x_1 + 2
POL( U41_3(x_1, ..., x_3) ) = 0
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U71_1(x_1) ) = x_1
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = x_1
POL( U94_2(x_1, x_2) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = 0
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U13_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = 2x_1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(isNat(X)) -> active(isNat(X))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U94(tt, L)) -> mark(s(length(L)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U61(X)) -> active(U61(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   mark(U85(X1, X2)) -> active(U85(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U86(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)


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

(216)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2x_1 + 2
POL( MARK_1(x_1) ) = 2x_1 + 2
POL( U31_2(x_1, x_2) ) = 1
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = 2x_1
POL( U61_1(x_1) ) = x_1
POL( U83_3(x_1, ..., x_3) ) = 0
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U86_1(x_1) ) = x_1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U71_1(x_1) ) = x_1
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = 2
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 2
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1
POL( isNat_1(x_1) ) = 0
POL( U33_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(isNat(X)) -> active(isNat(X))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U94(tt, L)) -> mark(s(length(L)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U61(X)) -> active(U61(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   mark(U85(X1, X2)) -> active(U85(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U86(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)


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

(218)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = x_1 + 1
POL( U83_3(x_1, ..., x_3) ) = 0
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1 + 2
POL( U86_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_2
POL( zeros ) = 2
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = x_2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U13_1(x_1) ) = 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U32_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)


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

(220)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(isNatKind(length(V1))) -> MARK(U61(isNatIListKind(V1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2x_1
POL( MARK_1(x_1) ) = 2x_1
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = x_1
POL( U61_1(x_1) ) = x_1
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( isNatIListKind_1(x_1) ) = 2x_1 + 2
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U86_1(x_1) ) = 2x_1
POL( U91_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U71_1(x_1) ) = x_1
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = x_2 + 2
POL( isNatKind_1(x_1) ) = 2x_1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( length_1(x_1) ) = x_1 + 2
POL( cons_2(x_1, x_2) ) = x_1 + 2x_2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 0
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 0
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(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)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(isNat(X)) -> active(isNat(X))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U94(tt, L)) -> mark(s(length(L)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U61(X)) -> active(U61(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   mark(U85(X1, X2)) -> active(U85(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U86(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)


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

(222)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U61(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U83_3(x_1, ..., x_3) ) = x_1 + 1
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = 2x_1 + 1
POL( U86_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = x_1 + 1
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = x_1 + 2
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)


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

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

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatKind(s(V1))) -> MARK(U71(isNatKind(V1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, x_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = 2x_1 + 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U86_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2
POL( U71_1(x_1) ) = x_1 + 2
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 2x_1 + 2
POL( U45_2(x_1, x_2) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2x_2 + 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = 2x_1 + 2
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 2
POL( U61_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)


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

(226)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

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

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( MARK_1(x_1) ) = 1
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( isNatKind_1(x_1) ) = 1
POL( U45_2(x_1, x_2) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = x_2 + 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 2
POL( U13_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U32_2(x_1, x_2) ) = 2x_1 + 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = 2
POL( U61_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(228)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U71(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2
POL( MARK_1(x_1) ) = x_1 + 2
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = 2x_1
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U86_1(x_1) ) = 2x_1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = x_1
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U71_1(x_1) ) = x_1 + 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U13_1(x_1) ) = x_1 + 2
POL( U23_1(x_1) ) = 2x_1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = 2x_1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(230)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = x_1 + 1
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = 0
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = x_1 + 1
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = 2x_1
POL( U86_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = x_2 + 2
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2x_1 + x_2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = x_1 + 1
POL( U33_1(x_1) ) = x_1 + 2
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(232)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

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

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( MARK_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = 0
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( U71_1(x_1) ) = 1
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = x_2 + 1
POL( U13_1(x_1) ) = x_1
POL( U23_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = 2x_1
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U61_1(x_1) ) = 2x_1 + 2
POL( nil ) = 0

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

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(234)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U86_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 1
POL( mark_1(x_1) ) = 2x_1
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = 1
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U45_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = 1
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U71_1(x_1) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
POL( 0 ) = 1
POL( U11_2(x_1, x_2) ) = 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2x_2 + 2
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = x_1 + 1
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = 2
POL( nil ) = 1

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(236)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = max{0, x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 1
POL( U51_2(x_1, x_2) ) = 2
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 1
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_1(x_1) ) = x_1 + 2
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(238)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U52(X)) -> MARK(X)
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U52(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U86_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 1
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2x_1 + 2
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 1}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = x_1 + 2
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(240)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = 1
POL( U41_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U81_3(x_1, ..., x_3) ) = 0
POL( U82_3(x_1, ..., x_3) ) = 0
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U45_2(x_1, x_2) ) = x_2 + 2
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = 1
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = 0
POL( U51_2(x_1, x_2) ) = 2
POL( U71_1(x_1) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2x_1 + x_2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 2x_2 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = x_1 + 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( U52_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 1

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(242)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

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

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( MARK_1(x_1) ) = 1
POL( U86_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( mark_1(x_1) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( U51_2(x_1, x_2) ) = 2x_2 + 2
POL( U71_1(x_1) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( zeros ) = 2
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(244)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = 0
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 0
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U71_1(x_1) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(246)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = 0
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = 0
POL( U85_2(x_1, x_2) ) = 0
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( zeros ) = 2
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U33_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = 0
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(248)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 0
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = 2x_1 + 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = x_1 + 2
POL( nil ) = 2

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(250)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 0
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = 2
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = x_1 + 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2x_1
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(252)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = 2x_1
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = 0
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = 1
POL( U84_3(x_1, ..., x_3) ) = 1
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 1}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = 2
POL( nil ) = 2

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(254)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = 0
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = 2x_1 + x_2
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = x_1 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = x_1 + 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2x_1
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = x_1
POL( U52_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(256)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 1
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = x_1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U81_3(x_1, ..., x_3) ) = 1
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = 0
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( U42_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U71_1(x_1) ) = 1
POL( isNatKind_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 1
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U32_2(x_1, x_2) ) = 1
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(258)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_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( U94_2(x_1, x_2) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = x_1
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(260)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2
POL( MARK_1(x_1) ) = x_1 + 2
POL( U86_1(x_1) ) = x_1
POL( U91_3(x_1, ..., x_3) ) = 2x_1
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = x_1 + 2
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = x_2 + 1
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = x_1 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U51_2(x_1, x_2) ) = 2x_2 + 2
POL( U71_1(x_1) ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U13_1(x_1) ) = x_1 + 2
POL( U23_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U46_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2x_1 + 2
POL( s_1(x_1) ) = 2x_1
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(262)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = x_1 + 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_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( U94_2(x_1, x_2) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = 0
POL( U85_2(x_1, x_2) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 - 1}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_1
POL( zeros ) = 0
POL( 0 ) = 1
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 0
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U45_2(x_1, x_2) ) = 2x_1 + 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(264)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U46(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 0
POL( U81_3(x_1, ..., x_3) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = 0
POL( U94_2(x_1, x_2) ) = 0
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = 1
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U71_1(x_1) ) = x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U45_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U46_1(x_1) ) = x_1 + 2
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(266)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U81(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U86_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatList_1(x_1) ) = x_1 + 1
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = 1
POL( U81_3(x_1, ..., x_3) ) = x_3 + 2
POL( U82_3(x_1, ..., x_3) ) = x_3 + 2
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = x_3 + 2
POL( U84_3(x_1, ..., x_3) ) = x_3 + 2
POL( U85_2(x_1, x_2) ) = x_2 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = x_1 + x_2
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_2 + 2
POL( zeros ) = 2
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2x_1 + 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(268)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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(isNatList(X)) -> ACTIVE(isNatList(X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( MARK_1(x_1) ) = 2
POL( U86_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 1
POL( zeros ) = 1
POL( 0 ) = 1
POL( U11_2(x_1, x_2) ) = 2x_1 + 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = x_1 + x_2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = x_1 + x_3 + 2
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = 2
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(270)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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(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( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 1
POL( U82_3(x_1, ..., x_3) ) = 0
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = x_1
POL( U81_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U83_3(x_1, ..., x_3) ) = 1
POL( U84_3(x_1, ..., x_3) ) = 0
POL( U85_2(x_1, x_2) ) = 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( length_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 - 1}
POL( U51_2(x_1, x_2) ) = x_2 + 2
POL( U71_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( zeros ) = 0
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = x_2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 2x_1 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U45_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( U86_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(272)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U81(tt, V1, V2)) -> MARK(U82(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( mark_1(x_1) ) = 0
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 1
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = x_1
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U45_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = x_1 + 2
POL( isNatIList_1(x_1) ) = 2
POL( U52_1(x_1) ) = 2
POL( U86_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = max{0, 2x_1 - 1}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(274)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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(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( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = x_1 + 2
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2
POL( U83_3(x_1, ..., x_3) ) = max{0, -2}
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( U42_3(x_1, ..., x_3) ) = x_2 + 2
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 1}
POL( isNatKind_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 1}
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 1
POL( U52_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(276)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U82(tt, V1, V2)) -> MARK(U83(isNatIListKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = 1
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( length_1(x_1) ) = 1
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 + 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = 0
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U45_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 0
POL( U86_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = 2
POL( U61_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(278)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 0
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = 2x_1
POL( U83_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U84_3(x_1, ..., x_3) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( zeros ) = 1
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = x_1 + 2
POL( U33_1(x_1) ) = 0
POL( U43_3(x_1, ..., x_3) ) = x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U45_2(x_1, x_2) ) = x_2 + 2
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = x_1
POL( U52_1(x_1) ) = max{0, -2}
POL( U86_1(x_1) ) = x_1 + 1
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(280)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U83(tt, V1, V2)) -> MARK(U84(isNatIListKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 2
POL( U94_2(x_1, x_2) ) = 2
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = 0
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_1 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 2x_1
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( isNatList_1(x_1) ) = 2x_1
POL( U23_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2x_1
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U86_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 1

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(282)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = 2x_1
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U85_2(x_1, x_2) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = x_2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U33_1(x_1) ) = 1
POL( U43_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( U52_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = 2x_1 + 1
POL( U61_1(x_1) ) = 2x_1 + 2
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(284)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U86(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U86(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = x_1
POL( U21_2(x_1, x_2) ) = x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U94_2(x_1, x_2) ) = 2x_1 + 1
POL( ACTIVE_1(x_1) ) = 1
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U85_2(x_1, x_2) ) = 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( length_1(x_1) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, x_3 - 1}
POL( U42_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = max{0, x_3 - 1}
POL( U82_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = x_2 + 2
POL( U33_1(x_1) ) = x_1 + 2
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U44_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2
POL( U45_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U83_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U84_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U86_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = x_1
POL( U61_1(x_1) ) = 0
POL( nil ) = 0

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

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(286)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

   ACTIVE(U84(tt, V1, V2)) -> MARK(U85(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U85_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U82_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 0
POL( U43_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U86_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = x_1
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(288)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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(U85(X1, X2)) -> ACTIVE(U85(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U85_2(x_1, x_2) ) = x_2 + 2
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = x_1 + 2
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U82_3(x_1, ..., x_3) ) = x_2 + x_3 + 1
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 1
POL( 0 ) = 2
POL( U11_2(x_1, x_2) ) = 2x_1 + 2
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = x_2 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2x_1
POL( U52_1(x_1) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U84_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U86_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = x_1 + 1
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(290)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

   ACTIVE(U85(tt, V2)) -> MARK(U86(isNatList(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = 2
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2
POL( U82_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2x_1 + x_2 + 1
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2x_1
POL( U52_1(x_1) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = x_1 + 2
POL( U86_1(x_1) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(292)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

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

POL( MARK_1(x_1) ) = 2x_1
POL( U21_2(x_1, x_2) ) = 2x_1 + x_2
POL( U22_2(x_1, x_2) ) = x_2 + 2
POL( U94_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( ACTIVE_1(x_1) ) = 2x_1
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 2x_2
POL( U92_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U93_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( length_1(x_1) ) = 2x_1
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = 1
POL( U71_1(x_1) ) = 1
POL( U81_3(x_1, ..., x_3) ) = 2x_3
POL( U82_3(x_1, ..., x_3) ) = 2x_3
POL( isNatKind_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = 2x_2
POL( zeros ) = 0
POL( 0 ) = 1
POL( U11_2(x_1, x_2) ) = 2x_2
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = 2x_2
POL( isNatIListKind_1(x_1) ) = 2
POL( U13_1(x_1) ) = x_1
POL( isNatList_1(x_1) ) = x_1
POL( U23_1(x_1) ) = x_1 + 2
POL( isNat_1(x_1) ) = x_1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 1
POL( U33_1(x_1) ) = 1
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( U52_1(x_1) ) = 1
POL( U83_3(x_1, ..., x_3) ) = 2x_3
POL( U84_3(x_1, ..., x_3) ) = 2x_3
POL( U85_2(x_1, x_2) ) = 2x_2
POL( U86_1(x_1) ) = 2x_1
POL( s_1(x_1) ) = x_1 + 2
POL( U61_1(x_1) ) = 1
POL( nil ) = 1

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

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(isNat(X)) -> active(isNat(X))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U94(tt, L)) -> mark(s(length(L)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U61(X)) -> active(U61(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3))
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3))
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   mark(U85(X1, X2)) -> active(U85(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U86(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   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)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   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)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(active(X)) -> U86(X)
   U86(mark(X)) -> U86(X)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   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)


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

(294)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = 0
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U71_1(x_1) ) = 2x_1
POL( U81_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = 1
POL( isNat_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2x_1 + 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = x_1 + 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U83_3(x_1, ..., x_3) ) = 2
POL( U84_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U85_2(x_1, x_2) ) = 2x_1 + 2
POL( U86_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 2

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(296)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(length(cons(N, L))) -> MARK(U91(isNatList(L), L, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = 1
POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 0
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U82_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U13_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2x_1 + 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = 2x_2 + 2
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U84_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( U85_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U86_1(x_1) ) = 2
POL( s_1(x_1) ) = 2
POL( length_1(x_1) ) = 2
POL( U61_1(x_1) ) = 0
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


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

(298)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U22_2(x_1, x_2) ) = 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = 2
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U51_2(x_1, x_2) ) = 2
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = x_2 + 2
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 1
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U45_2(x_1, x_2) ) = x_2 + 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 2x_1
POL( U52_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U86_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U61_1(x_1) ) = 2
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)


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

(300)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U94(tt, L)) -> MARK(s(length(L)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, 2x_1 - 1}
POL( U22_2(x_1, x_2) ) = 1
POL( U94_2(x_1, x_2) ) = 1
POL( ACTIVE_1(x_1) ) = 1
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U21_2(x_1, x_2) ) = 2x_2 + 2
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2
POL( U82_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2
POL( isNatKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = 2x_1 + 2
POL( U85_2(x_1, x_2) ) = x_2 + 2
POL( isNat_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( tt ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = 2x_2 + 2
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = x_1
POL( U52_1(x_1) ) = 2x_1 + 2
POL( U83_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2
POL( U84_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U86_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( s_1(x_1) ) = 0
POL( length_1(x_1) ) = 2
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> 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)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)


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

(302)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

   ACTIVE(U91(tt, L, N)) -> MARK(U92(isNatIListKind(L), L, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U82_3(x_1, ..., x_3) ) = x_2 + x_3
POL( isNatKind_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( tt ) = 1
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U84_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U85_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( U86_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( s_1(x_1) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( U61_1(x_1) ) = 0
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)


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

(304)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U22_2(x_1, x_2) ) = 1
POL( U94_2(x_1, x_2) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U21_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( U82_3(x_1, ..., x_3) ) = x_1 + x_3 + 1
POL( isNatKind_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( tt ) = 0
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U13_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2
POL( U83_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U84_3(x_1, ..., x_3) ) = 2
POL( U85_2(x_1, x_2) ) = 2
POL( U86_1(x_1) ) = max{0, x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( s_1(x_1) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( U61_1(x_1) ) = max{0, -2}
POL( nil ) = 0

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)


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

(306)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

Q is empty.
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.

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U92(tt, L, N)) -> MARK(U93(isNat(N), L, N))
   ACTIVE(U93(tt, L, N)) -> MARK(U94(isNatKind(N), L))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  x1

U21(x1, x2)  =  U21

MARK(x1)  =  x1

U22(x1, x2)  =  U22

U92(x1, x2, x3)  =  U92

U93(x1, x2, x3)  =  U93

U94(x1, x2)  =  U94


Knuth-Bendix order [KBO] with precedence:U93 > U94

and weight map:

   U93=1
   U92=2
   U21=3
   U22=2
   U94=1

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

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)


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

(308)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2)) -> ACTIVE(U94(mark(X1), X2))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U13(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, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt, V1, V2)) -> mark(U82(isNatKind(V1), V1, V2))
   active(U82(tt, V1, V2)) -> mark(U83(isNatIListKind(V2), V1, V2))
   active(U83(tt, V1, V2)) -> mark(U84(isNatIListKind(V2), V1, V2))
   active(U84(tt, V1, V2)) -> mark(U85(isNat(V1), V2))
   active(U85(tt, V2)) -> mark(U86(isNatList(V2)))
   active(U86(tt)) -> mark(tt)
   active(U91(tt, L, N)) -> mark(U92(isNatIListKind(L), L, N))
   active(U92(tt, L, N)) -> mark(U93(isNat(N), L, N))
   active(U93(tt, L, N)) -> mark(U94(isNatKind(N), L))
   active(U94(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U61(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U71(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U81(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U91(isNatList(L), L, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   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(U85(X1, X2)) -> active(U85(mark(X1), X2))
   mark(U86(X)) -> active(U86(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2)) -> active(U94(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(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)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   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)
   U85(mark(X1), X2) -> U85(X1, X2)
   U85(X1, mark(X2)) -> U85(X1, X2)
   U85(active(X1), X2) -> U85(X1, X2)
   U85(X1, active(X2)) -> U85(X1, X2)
   U86(mark(X)) -> U86(X)
   U86(active(X)) -> U86(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2) -> U94(X1, X2)
   U94(X1, mark(X2)) -> U94(X1, X2)
   U94(active(X1), X2) -> U94(X1, X2)
   U94(X1, active(X2)) -> U94(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)

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

(309) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
----------------------------------------

(310)
TRUE