YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 191 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 8 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QReductionProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) UsableRulesProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QReductionProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) UsableRulesProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QReductionProof [EQUIVALENT, 0 ms]
        (23) QDP
        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (25) YES
    (26) QDP
        (27) UsableRulesProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QReductionProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (32) YES
    (33) QDP
        (34) UsableRulesProof [EQUIVALENT, 0 ms]
        (35) QDP
        (36) QReductionProof [EQUIVALENT, 0 ms]
        (37) QDP
        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (39) YES
    (40) QDP
        (41) UsableRulesProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QReductionProof [EQUIVALENT, 1 ms]
        (44) QDP
        (45) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (46) YES
    (47) QDP
        (48) UsableRulesProof [EQUIVALENT, 0 ms]
        (49) QDP
        (50) QReductionProof [EQUIVALENT, 0 ms]
        (51) QDP
        (52) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (53) YES
    (54) QDP
        (55) UsableRulesProof [EQUIVALENT, 0 ms]
        (56) QDP
        (57) QReductionProof [EQUIVALENT, 0 ms]
        (58) QDP
        (59) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (60) YES
    (61) QDP
        (62) UsableRulesProof [EQUIVALENT, 0 ms]
        (63) QDP
        (64) QReductionProof [EQUIVALENT, 1 ms]
        (65) QDP
        (66) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (67) YES
    (68) QDP
        (69) UsableRulesProof [EQUIVALENT, 0 ms]
        (70) QDP
        (71) QReductionProof [EQUIVALENT, 0 ms]
        (72) QDP
        (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (74) YES
    (75) QDP
        (76) UsableRulesProof [EQUIVALENT, 0 ms]
        (77) QDP
        (78) QReductionProof [EQUIVALENT, 0 ms]
        (79) QDP
        (80) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (81) YES
    (82) QDP
        (83) UsableRulesProof [EQUIVALENT, 0 ms]
        (84) QDP
        (85) QReductionProof [EQUIVALENT, 1 ms]
        (86) QDP
        (87) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (88) YES
    (89) QDP
        (90) UsableRulesProof [EQUIVALENT, 0 ms]
        (91) QDP
        (92) QReductionProof [EQUIVALENT, 2 ms]
        (93) QDP
        (94) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (95) YES
    (96) QDP
        (97) UsableRulesProof [EQUIVALENT, 0 ms]
        (98) QDP
        (99) QReductionProof [EQUIVALENT, 0 ms]
        (100) QDP
        (101) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (102) YES
    (103) QDP
        (104) UsableRulesProof [EQUIVALENT, 0 ms]
        (105) QDP
        (106) QReductionProof [EQUIVALENT, 0 ms]
        (107) QDP
        (108) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (109) YES
    (110) QDP
        (111) UsableRulesProof [EQUIVALENT, 0 ms]
        (112) QDP
        (113) QReductionProof [EQUIVALENT, 0 ms]
        (114) QDP
        (115) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (116) YES
    (117) QDP
        (118) UsableRulesProof [EQUIVALENT, 0 ms]
        (119) QDP
        (120) QReductionProof [EQUIVALENT, 0 ms]
        (121) QDP
        (122) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (123) YES
    (124) QDP
        (125) UsableRulesProof [EQUIVALENT, 0 ms]
        (126) QDP
        (127) QReductionProof [EQUIVALENT, 0 ms]
        (128) QDP
        (129) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (130) YES
    (131) QDP
        (132) UsableRulesProof [EQUIVALENT, 0 ms]
        (133) QDP
        (134) QReductionProof [EQUIVALENT, 0 ms]
        (135) QDP
        (136) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (137) YES
    (138) QDP
        (139) UsableRulesProof [EQUIVALENT, 0 ms]
        (140) QDP
        (141) QReductionProof [EQUIVALENT, 0 ms]
        (142) QDP
        (143) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (144) YES
    (145) QDP
        (146) UsableRulesProof [EQUIVALENT, 0 ms]
        (147) QDP
        (148) QReductionProof [EQUIVALENT, 0 ms]
        (149) QDP
        (150) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (151) YES
    (152) QDP
        (153) UsableRulesProof [EQUIVALENT, 0 ms]
        (154) QDP
        (155) QReductionProof [EQUIVALENT, 0 ms]
        (156) QDP
        (157) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (158) YES
    (159) QDP
        (160) UsableRulesProof [EQUIVALENT, 0 ms]
        (161) QDP
        (162) QDPOrderProof [EQUIVALENT, 875 ms]
        (163) QDP
        (164) QDPOrderProof [EQUIVALENT, 70 ms]
        (165) QDP
        (166) QDPOrderProof [EQUIVALENT, 816 ms]
        (167) QDP
        (168) QDPQMonotonicMRRProof [EQUIVALENT, 434 ms]
        (169) QDP
        (170) QDPQMonotonicMRRProof [EQUIVALENT, 485 ms]
        (171) QDP
        (172) DependencyGraphProof [EQUIVALENT, 0 ms]
        (173) QDP
        (174) QDPOrderProof [EQUIVALENT, 905 ms]
        (175) QDP
        (176) DependencyGraphProof [EQUIVALENT, 0 ms]
        (177) QDP
        (178) QDPOrderProof [EQUIVALENT, 593 ms]
        (179) QDP
        (180) DependencyGraphProof [EQUIVALENT, 0 ms]
        (181) QDP
        (182) QDPOrderProof [EQUIVALENT, 584 ms]
        (183) QDP
        (184) DependencyGraphProof [EQUIVALENT, 0 ms]
        (185) QDP
        (186) QDPOrderProof [EQUIVALENT, 518 ms]
        (187) QDP
        (188) QDPOrderProof [EQUIVALENT, 984 ms]
        (189) QDP
        (190) DependencyGraphProof [EQUIVALENT, 0 ms]
        (191) QDP
        (192) QDPOrderProof [EQUIVALENT, 915 ms]
        (193) QDP
        (194) QDPOrderProof [EQUIVALENT, 587 ms]
        (195) QDP
        (196) DependencyGraphProof [EQUIVALENT, 0 ms]
        (197) QDP
        (198) QDPQMonotonicMRRProof [EQUIVALENT, 419 ms]
        (199) QDP
        (200) QDPQMonotonicMRRProof [EQUIVALENT, 414 ms]
        (201) QDP
        (202) QDPQMonotonicMRRProof [EQUIVALENT, 369 ms]
        (203) QDP
        (204) QDPQMonotonicMRRProof [EQUIVALENT, 245 ms]
        (205) QDP
        (206) QDPQMonotonicMRRProof [EQUIVALENT, 319 ms]
        (207) QDP
        (208) QDPOrderProof [EQUIVALENT, 420 ms]
        (209) QDP
        (210) QDPOrderProof [EQUIVALENT, 1012 ms]
        (211) QDP
        (212) DependencyGraphProof [EQUIVALENT, 0 ms]
        (213) QDP
        (214) QDPOrderProof [EQUIVALENT, 400 ms]
        (215) QDP
        (216) QDPOrderProof [EQUIVALENT, 396 ms]
        (217) QDP
        (218) QDPQMonotonicMRRProof [EQUIVALENT, 274 ms]
        (219) QDP
        (220) QDPQMonotonicMRRProof [EQUIVALENT, 371 ms]
        (221) QDP
        (222) QDPOrderProof [EQUIVALENT, 508 ms]
        (223) QDP
        (224) QDPOrderProof [EQUIVALENT, 727 ms]
        (225) QDP
        (226) DependencyGraphProof [EQUIVALENT, 0 ms]
        (227) QDP
        (228) QDPOrderProof [EQUIVALENT, 373 ms]
        (229) QDP
        (230) QDPOrderProof [EQUIVALENT, 310 ms]
        (231) QDP
        (232) QDPOrderProof [EQUIVALENT, 1005 ms]
        (233) QDP
        (234) QDPOrderProof [EQUIVALENT, 1625 ms]
        (235) QDP
        (236) DependencyGraphProof [EQUIVALENT, 0 ms]
        (237) QDP
        (238) TransformationProof [EQUIVALENT, 0 ms]
        (239) QDP
        (240) QDPOrderProof [EQUIVALENT, 0 ms]
        (241) QDP
        (242) DependencyGraphProof [EQUIVALENT, 0 ms]
        (243) TRUE


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

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

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

(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)) -> MARK(tt)
   ACTIVE(U21(tt)) -> MARK(tt)
   ACTIVE(U31(tt)) -> MARK(tt)
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U41(tt, V2)) -> U42^1(isNatIList(V2))
   ACTIVE(U41(tt, V2)) -> ISNATILIST(V2)
   ACTIVE(U42(tt)) -> MARK(tt)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   ACTIVE(U51(tt, V2)) -> U52^1(isNatList(V2))
   ACTIVE(U51(tt, V2)) -> ISNATLIST(V2)
   ACTIVE(U52(tt)) -> MARK(tt)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   ACTIVE(U61(tt, V2)) -> U62^1(isNatIList(V2))
   ACTIVE(U61(tt, V2)) -> ISNATILIST(V2)
   ACTIVE(U62(tt)) -> MARK(tt)
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   ACTIVE(U71(tt, L, N)) -> U72^1(isNat(N), L)
   ACTIVE(U71(tt, L, N)) -> ISNAT(N)
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U72(tt, L)) -> S(length(L))
   ACTIVE(U72(tt, L)) -> LENGTH(L)
   ACTIVE(U81(tt)) -> MARK(nil)
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   ACTIVE(U91(tt, IL, M, N)) -> U92^1(isNat(M), IL, M, N)
   ACTIVE(U91(tt, IL, M, N)) -> ISNAT(M)
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   ACTIVE(U92(tt, IL, M, N)) -> U93^1(isNat(N), IL, M, N)
   ACTIVE(U92(tt, IL, M, N)) -> ISNAT(N)
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U93(tt, IL, M, N)) -> CONS(N, take(M, IL))
   ACTIVE(U93(tt, IL, M, N)) -> TAKE(M, IL)
   ACTIVE(isNat(0)) -> MARK(tt)
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(length(V1))) -> U11^1(isNatList(V1))
   ACTIVE(isNat(length(V1))) -> ISNATLIST(V1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   ACTIVE(isNat(s(V1))) -> U21^1(isNat(V1))
   ACTIVE(isNat(s(V1))) -> ISNAT(V1)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(V)) -> U31^1(isNatList(V))
   ACTIVE(isNatIList(V)) -> ISNATLIST(V)
   ACTIVE(isNatIList(zeros)) -> MARK(tt)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> U41^1(isNat(V1), V2)
   ACTIVE(isNatIList(cons(V1, V2))) -> ISNAT(V1)
   ACTIVE(isNatList(nil)) -> MARK(tt)
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(cons(V1, V2))) -> U51^1(isNat(V1), V2)
   ACTIVE(isNatList(cons(V1, V2))) -> ISNAT(V1)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> U61^1(isNat(V1), V2)
   ACTIVE(isNatList(take(V1, V2))) -> ISNAT(V1)
   ACTIVE(length(nil)) -> MARK(0)
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   ACTIVE(length(cons(N, L))) -> U71^1(isNatList(L), L, N)
   ACTIVE(length(cons(N, L))) -> ISNATLIST(L)
   ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL)))
   ACTIVE(take(0, IL)) -> U81^1(isNatIList(IL))
   ACTIVE(take(0, IL)) -> ISNATILIST(IL)
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N))
   ACTIVE(take(s(M), cons(N, IL))) -> U91^1(isNatIList(IL), IL, M, N)
   ACTIVE(take(s(M), cons(N, IL))) -> ISNATILIST(IL)
   MARK(zeros) -> ACTIVE(zeros)
   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(cons(X1, X2)) -> CONS(mark(X1), X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(0) -> ACTIVE(0)
   MARK(U11(X)) -> ACTIVE(U11(mark(X)))
   MARK(U11(X)) -> U11^1(mark(X))
   MARK(U11(X)) -> MARK(X)
   MARK(tt) -> ACTIVE(tt)
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   MARK(U21(X)) -> U21^1(mark(X))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> ACTIVE(U31(mark(X)))
   MARK(U31(X)) -> U31^1(mark(X))
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> U41^1(mark(X1), X2)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   MARK(U42(X)) -> U42^1(mark(X))
   MARK(U42(X)) -> MARK(X)
   MARK(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(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> U61^1(mark(X1), X2)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   MARK(U62(X)) -> U62^1(mark(X))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U71(X1, X2, X3)) -> U71^1(mark(X1), X2, X3)
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   MARK(U72(X1, X2)) -> U72^1(mark(X1), X2)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   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(U81(X)) -> ACTIVE(U81(mark(X)))
   MARK(U81(X)) -> U81^1(mark(X))
   MARK(U81(X)) -> MARK(X)
   MARK(nil) -> ACTIVE(nil)
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> U91^1(mark(X1), X2, X3, X4)
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   MARK(U92(X1, X2, X3, X4)) -> U92^1(mark(X1), X2, X3, X4)
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   MARK(U93(X1, X2, X3, X4)) -> U93^1(mark(X1), X2, X3, X4)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> TAKE(mark(X1), mark(X2))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   CONS(mark(X1), X2) -> 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(X)) -> U11^1(X)
   U11^1(active(X)) -> U11^1(X)
   U21^1(mark(X)) -> U21^1(X)
   U21^1(active(X)) -> U21^1(X)
   U31^1(mark(X)) -> U31^1(X)
   U31^1(active(X)) -> U31^1(X)
   U41^1(mark(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, mark(X2)) -> U41^1(X1, X2)
   U41^1(active(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, active(X2)) -> U41^1(X1, X2)
   U42^1(mark(X)) -> U42^1(X)
   U42^1(active(X)) -> U42^1(X)
   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)
   ISNATLIST(mark(X)) -> ISNATLIST(X)
   ISNATLIST(active(X)) -> ISNATLIST(X)
   U61^1(mark(X1), X2) -> U61^1(X1, X2)
   U61^1(X1, mark(X2)) -> U61^1(X1, X2)
   U61^1(active(X1), X2) -> U61^1(X1, X2)
   U61^1(X1, active(X2)) -> U61^1(X1, X2)
   U62^1(mark(X)) -> U62^1(X)
   U62^1(active(X)) -> U62^1(X)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)
   U72^1(mark(X1), X2) -> U72^1(X1, X2)
   U72^1(X1, mark(X2)) -> U72^1(X1, X2)
   U72^1(active(X1), X2) -> U72^1(X1, X2)
   U72^1(X1, active(X2)) -> U72^1(X1, X2)
   ISNAT(mark(X)) -> ISNAT(X)
   ISNAT(active(X)) -> ISNAT(X)
   S(mark(X)) -> S(X)
   S(active(X)) -> S(X)
   LENGTH(mark(X)) -> LENGTH(X)
   LENGTH(active(X)) -> LENGTH(X)
   U81^1(mark(X)) -> U81^1(X)
   U81^1(active(X)) -> U81^1(X)
   U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4)
   U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4)
   U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4)
   U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4)
   U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4)
   U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4)
   TAKE(mark(X1), X2) -> TAKE(X1, X2)
   TAKE(X1, mark(X2)) -> TAKE(X1, X2)
   TAKE(active(X1), X2) -> TAKE(X1, X2)
   TAKE(X1, active(X2)) -> TAKE(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 23 SCCs with 68 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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


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


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


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

(11)
YES

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

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

   U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4)
   U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4)
   U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

   U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4)
   U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4)
   U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

From the DPs we obtained the following set of size-change graphs:
*U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


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

(18)
YES

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

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

   U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4)
   U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4)
   U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

   U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4)
   U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4)
   U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

From the DPs we obtained the following set of size-change graphs:
*U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


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

(25)
YES

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

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

   U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4)
   U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4)
   U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

   U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4)
   U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4)
   U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

From the DPs we obtained the following set of size-change graphs:
*U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


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

(32)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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


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

(39)
YES

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

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

   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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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

(46)
YES

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

(47)
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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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

(53)
YES

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

(54)
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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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

(60)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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


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


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


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

(67)
YES

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

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

   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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:
*U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(74)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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


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

(81)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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


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


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


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

(88)
YES

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

(89)
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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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

(95)
YES

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

(96)
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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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

(102)
YES

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

(103)
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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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:
*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


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

(109)
YES

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

(110)
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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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

(116)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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


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

(123)
YES

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

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

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

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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


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


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


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

(130)
YES

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

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

   U31^1(active(X)) -> U31^1(X)
   U31^1(mark(X)) -> U31^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   U31^1(active(X)) -> U31^1(X)
   U31^1(mark(X)) -> U31^1(X)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

   U31^1(active(X)) -> U31^1(X)
   U31^1(mark(X)) -> U31^1(X)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

From the DPs we obtained the following set of size-change graphs:
*U31^1(active(X)) -> U31^1(X)
The graph contains the following edges 1 > 1


*U31^1(mark(X)) -> U31^1(X)
The graph contains the following edges 1 > 1


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

(137)
YES

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

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

   U21^1(active(X)) -> U21^1(X)
   U21^1(mark(X)) -> U21^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   U21^1(active(X)) -> U21^1(X)
   U21^1(mark(X)) -> U21^1(X)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

   U21^1(active(X)) -> U21^1(X)
   U21^1(mark(X)) -> U21^1(X)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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:
*U21^1(active(X)) -> U21^1(X)
The graph contains the following edges 1 > 1


*U21^1(mark(X)) -> U21^1(X)
The graph contains the following edges 1 > 1


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

(144)
YES

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

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

   U11^1(active(X)) -> U11^1(X)
   U11^1(mark(X)) -> U11^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   U11^1(active(X)) -> U11^1(X)
   U11^1(mark(X)) -> U11^1(X)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

   U11^1(active(X)) -> U11^1(X)
   U11^1(mark(X)) -> U11^1(X)

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

From the DPs we obtained the following set of size-change graphs:
*U11^1(active(X)) -> U11^1(X)
The graph contains the following edges 1 > 1


*U11^1(mark(X)) -> U11^1(X)
The graph contains the following edges 1 > 1


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

(151)
YES

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

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

   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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))


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

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

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

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

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))

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

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

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


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

(158)
YES

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

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

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X)) -> ACTIVE(U11(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL)))
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> ACTIVE(U31(mark(X)))
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   active(U72(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X)) -> active(U11(mark(X)))
   mark(tt) -> active(tt)
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X)) -> active(U31(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U42(X)) -> active(U42(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(isNatList(X)) -> active(isNatList(X))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(nil) -> active(nil)
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   cons(mark(X1), X2) -> 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(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   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)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X)) -> ACTIVE(U11(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL)))
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> ACTIVE(U31(mark(X)))
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(U42(X)) -> ACTIVE(U42(mark(X)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(U11(X)) -> ACTIVE(U11(mark(X)))
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   MARK(U31(X)) -> ACTIVE(U31(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_1(x_1) ) = max{0, -2}
POL( U21_1(x_1) ) = max{0, -2}
POL( U31_1(x_1) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = 2
POL( U42_1(x_1) ) = 0
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = 0
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_3(x_1, ..., x_3) ) = 2
POL( U72_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = max{0, -2}
POL( U91_4(x_1, ..., x_4) ) = 2
POL( U92_4(x_1, ..., x_4) ) = 2
POL( U93_4(x_1, ..., x_4) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( 0 ) = 0
POL( tt ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( isNat_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)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


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

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

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL)))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U72(X1, X2)) -> MARK(X1)
   MARK(length(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U11(x_1)) = x_1
   POL(U21(x_1)) = x_1
   POL(U31(x_1)) = x_1
   POL(U41(x_1, x_2)) = x_1
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1, x_2, x_3)) = 1 + x_1 + x_2
   POL(U72(x_1, x_2)) = 1 + x_1 + x_2
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(U92(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(U93(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 1 + x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = x_1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

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

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

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

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL)))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL)))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N))
   MARK(U81(X)) -> MARK(X)
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
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( U41_2(x_1, x_2) ) = x_1
POL( U51_2(x_1, x_2) ) = x_1
POL( U61_2(x_1, x_2) ) = 2x_1
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( U91_4(x_1, ..., x_4) ) = x_1 + x_3 + x_4
POL( U92_4(x_1, ..., x_4) ) = 2x_1 + x_4
POL( U93_4(x_1, ..., x_4) ) = 2x_1 + x_4
POL( length_1(x_1) ) = max{0, -2}
POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( mark_1(x_1) ) = x_1
POL( zeros ) = 0
POL( active_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = x_1
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = 2x_1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2x_1
POL( isNatList_1(x_1) ) = 0
POL( U62_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1
POL( U11_1(x_1) ) = 2x_1
POL( U21_1(x_1) ) = 2x_1
POL( U31_1(x_1) ) = 2x_1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2x_1

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

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

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

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(168) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = 1
   POL(U11(x_1)) = 0
   POL(U21(x_1)) = 0
   POL(U31(x_1)) = 0
   POL(U41(x_1, x_2)) = 1
   POL(U42(x_1)) = 0
   POL(U51(x_1, x_2)) = 1
   POL(U52(x_1)) = 0
   POL(U61(x_1, x_2)) = 1
   POL(U62(x_1)) = 0
   POL(U71(x_1, x_2, x_3)) = 1
   POL(U72(x_1, x_2)) = 1
   POL(U81(x_1)) = 0
   POL(U91(x_1, x_2, x_3, x_4)) = 1
   POL(U92(x_1, x_2, x_3, x_4)) = 1
   POL(U93(x_1, x_2, x_3, x_4)) = 1
   POL(active(x_1)) = 0
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = 1
   POL(isNatIList(x_1)) = 1
   POL(isNatList(x_1)) = 1
   POL(length(x_1)) = 1
   POL(mark(x_1)) = 0
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 1


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

(169)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(170) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   ACTIVE(zeros) -> MARK(cons(0, zeros))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2*x_1
   POL(U11(x_1)) = 2*x_1
   POL(U21(x_1)) = x_1
   POL(U31(x_1)) = x_1
   POL(U41(x_1, x_2)) = 2*x_1
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = 2*x_1
   POL(U52(x_1)) = 2*x_1
   POL(U61(x_1, x_2)) = 2*x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1, x_2, x_3)) = 0
   POL(U72(x_1, x_2)) = 0
   POL(U81(x_1)) = 0
   POL(U91(x_1, x_2, x_3, x_4)) = 2*x_1 + x_4
   POL(U92(x_1, x_2, x_3, x_4)) = 2*x_1 + x_4
   POL(U93(x_1, x_2, x_3, x_4)) = 2*x_1 + x_4
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 2 + 2*x_2
   POL(tt) = 0
   POL(zeros) = 2


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

(171)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   MARK(U42(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(U93(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U91(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U92(X1, X2, X3, X4)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1 + 2
POL( U41_2(x_1, x_2) ) = 2x_1
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U61_2(x_1, x_2) ) = 2x_1
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( U91_4(x_1, ..., x_4) ) = 2x_1 + 2x_4 + 2
POL( U92_4(x_1, ..., x_4) ) = x_1 + 2x_4 + 2
POL( U93_4(x_1, ..., x_4) ) = 2x_1 + 2x_4 + 2
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = x_1
POL( zeros ) = 2
POL( active_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = x_1 + 2
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = 2x_1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2x_1
POL( isNatList_1(x_1) ) = 0
POL( U62_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1
POL( take_2(x_1, x_2) ) = 2x_2
POL( U11_1(x_1) ) = x_1
POL( U21_1(x_1) ) = 2x_1
POL( U31_1(x_1) ) = x_1
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = x_1 + 2

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

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


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

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

   MARK(U42(X)) -> MARK(X)
   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 1}
POL( U41_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( U71_3(x_1, ..., x_3) ) = 1
POL( U72_2(x_1, x_2) ) = 1
POL( U91_4(x_1, ..., x_4) ) = 1
POL( U92_4(x_1, ..., x_4) ) = 1
POL( U93_4(x_1, ..., x_4) ) = max{0, -2}
POL( length_1(x_1) ) = 1
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 1
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( isNatList_1(x_1) ) = 1
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 1
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U11_1(x_1) ) = max{0, -2}
POL( U21_1(x_1) ) = max{0, 2x_1 - 2}
POL( U31_1(x_1) ) = max{0, x_1 - 2}
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

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

   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)


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

(179)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

(181)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U21(X)) -> MARK(X)
   MARK(U11(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(182) 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, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( U41_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U71_3(x_1, ..., x_3) ) = 2
POL( U72_2(x_1, x_2) ) = 2
POL( U91_4(x_1, ..., x_4) ) = 2
POL( U92_4(x_1, ..., x_4) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 2
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( isNatList_1(x_1) ) = 2
POL( U62_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( U93_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U11_1(x_1) ) = max{0, -2}
POL( U21_1(x_1) ) = max{0, -2}
POL( U31_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2

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

   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)


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

(183)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U21(X)) -> MARK(X)
   MARK(U11(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = x_1 + 1
POL( U51_2(x_1, x_2) ) = x_1 + 1
POL( U61_2(x_1, x_2) ) = x_1 + 1
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( U91_4(x_1, ..., x_4) ) = 2
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 2x_1 + 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = 2x_1 + 2
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U62_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U92_4(x_1, ..., x_4) ) = 2
POL( U93_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U11_1(x_1) ) = 2x_1 + 1
POL( U21_1(x_1) ) = x_1 + 1
POL( U31_1(x_1) ) = x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

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

   MARK(U11(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(X)) -> MARK(X)
   MARK(U31(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1)))
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( U41_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U51_2(x_1, x_2) ) = x_1 + x_2
POL( U61_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U71_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U72_2(x_1, x_2) ) = 2x_2 + 2
POL( length_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( zeros ) = 0
POL( active_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( U52_1(x_1) ) = x_1
POL( isNatList_1(x_1) ) = x_1
POL( U62_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 2x_1
POL( s_1(x_1) ) = x_1
POL( U91_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2
POL( U92_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2
POL( U93_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U11_1(x_1) ) = x_1 + 2
POL( U21_1(x_1) ) = x_1
POL( U31_1(x_1) ) = 2x_1 + 2
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = x_1

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

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


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

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

   MARK(U21(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

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

   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

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

POL( ACTIVE_1(x_1) ) = x_1 + 2
POL( U41_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = x_1 + 2
POL( U61_2(x_1, x_2) ) = 2
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = x_1
POL( zeros ) = 2
POL( active_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = x_1 + 2
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 2
POL( U52_1(x_1) ) = x_1
POL( isNatList_1(x_1) ) = 2
POL( U62_1(x_1) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1
POL( U91_4(x_1, ..., x_4) ) = 2x_1 + x_4 + 2
POL( U92_4(x_1, ..., x_4) ) = x_4 + 2
POL( U93_4(x_1, ..., x_4) ) = x_4 + 2
POL( take_2(x_1, x_2) ) = 2x_2 + 2
POL( U11_1(x_1) ) = max{0, -2}
POL( U21_1(x_1) ) = x_1
POL( U31_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = x_1 + 2

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

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


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

(193)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(194) 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(isNat(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( U41_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U71_3(x_1, ..., x_3) ) = 2
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = x_1 + 2
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = x_1 + 2
POL( isNatList_1(x_1) ) = 0
POL( U62_1(x_1) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 2x_1 + 2
POL( s_1(x_1) ) = x_1 + 2
POL( U91_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + x_3 + x_4 + 2
POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U93_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = x_2 + 2
POL( U11_1(x_1) ) = 2
POL( U21_1(x_1) ) = x_1 + 2
POL( U31_1(x_1) ) = 0
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 2}

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

   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)


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

(195)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

(197)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(198) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2*x_1
   POL(U11(x_1)) = 0
   POL(U21(x_1)) = x_1
   POL(U31(x_1)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = 1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = 0
   POL(U62(x_1)) = 2*x_1
   POL(U71(x_1, x_2, x_3)) = 0
   POL(U72(x_1, x_2)) = 0
   POL(U81(x_1)) = 0
   POL(U91(x_1, x_2, x_3, x_4)) = 0
   POL(U92(x_1, x_2, x_3, x_4)) = 0
   POL(U93(x_1, x_2, x_3, x_4)) = 0
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 1
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = 2*x_1
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0


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

(199)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(200) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.


Strictly oriented rules of the TRS R:

   active(length(nil)) -> mark(0)

Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2*x_1
   POL(U11(x_1)) = x_1
   POL(U21(x_1)) = x_1
   POL(U31(x_1)) = 2*x_1
   POL(U41(x_1, x_2)) = 2*x_1
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = 2*x_1
   POL(U52(x_1)) = 2*x_1
   POL(U61(x_1, x_2)) = 2*x_1
   POL(U62(x_1)) = 2*x_1
   POL(U71(x_1, x_2, x_3)) = x_1 + 2*x_2
   POL(U72(x_1, x_2)) = x_1 + 2*x_2
   POL(U81(x_1)) = 2 + x_1
   POL(U91(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + x_4
   POL(U92(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + x_4
   POL(U93(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + x_4
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 2*x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 2
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 2 + x_1 + x_2
   POL(tt) = 0
   POL(zeros) = 0


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

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

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(202) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2)))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2*x_1
   POL(U11(x_1)) = 0
   POL(U21(x_1)) = x_1
   POL(U31(x_1)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = 2*x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = 1
   POL(U62(x_1)) = x_1
   POL(U71(x_1, x_2, x_3)) = 0
   POL(U72(x_1, x_2)) = 0
   POL(U81(x_1)) = 0
   POL(U91(x_1, x_2, x_3, x_4)) = 2 + x_2
   POL(U92(x_1, x_2, x_3, x_4)) = 2 + x_2
   POL(U93(x_1, x_2, x_3, x_4)) = 2 + x_2
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 2*x_1
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 2 + x_2
   POL(tt) = 0
   POL(zeros) = 0


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

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

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(204) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2
   POL(U11(x_1)) = 0
   POL(U21(x_1)) = 0
   POL(U31(x_1)) = 0
   POL(U41(x_1, x_2)) = 1
   POL(U42(x_1)) = 0
   POL(U51(x_1, x_2)) = 1
   POL(U52(x_1)) = 0
   POL(U61(x_1, x_2)) = 0
   POL(U62(x_1)) = 0
   POL(U71(x_1, x_2, x_3)) = 1
   POL(U72(x_1, x_2)) = 1
   POL(U81(x_1)) = 0
   POL(U91(x_1, x_2, x_3, x_4)) = 0
   POL(U92(x_1, x_2, x_3, x_4)) = 0
   POL(U93(x_1, x_2, x_3, x_4)) = 0
   POL(active(x_1)) = 0
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 1
   POL(isNatList(x_1)) = 1
   POL(length(x_1)) = 1
   POL(mark(x_1)) = 0
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0


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

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

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(206) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(U62(X)) -> MARK(X)


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2*x_1
   POL(U11(x_1)) = 0
   POL(U21(x_1)) = 2*x_1
   POL(U31(x_1)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = 1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = 1
   POL(U62(x_1)) = 1 + 2*x_1
   POL(U71(x_1, x_2, x_3)) = 0
   POL(U72(x_1, x_2)) = 0
   POL(U81(x_1)) = 0
   POL(U91(x_1, x_2, x_3, x_4)) = 0
   POL(U92(x_1, x_2, x_3, x_4)) = 0
   POL(U93(x_1, x_2, x_3, x_4)) = 0
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 1
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = 2*x_1
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0


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

(207)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U21(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U21(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( U41_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 1
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = max{0, 2x_1 - 1}
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 1}
POL( 0 ) = 1
POL( tt ) = 0
POL( U42_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = 2x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = 1
POL( isNat_1(x_1) ) = x_1
POL( s_1(x_1) ) = x_1 + 1
POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_4 - 2}
POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U93_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3
POL( take_2(x_1, x_2) ) = max{0, x_2 - 1}
POL( U11_1(x_1) ) = max{0, 2x_1 - 2}
POL( U21_1(x_1) ) = x_1 + 2
POL( U31_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 1
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(209)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(210) 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(U51(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 - 1}
POL( U41_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_2 + 2
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2x_1 + 2
POL( zeros ) = 2
POL( active_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( 0 ) = 1
POL( tt ) = 0
POL( U42_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = x_1
POL( s_1(x_1) ) = x_1 + 1
POL( U91_4(x_1, ..., x_4) ) = 2x_1 + x_4 + 2
POL( U92_4(x_1, ..., x_4) ) = x_1 + 2
POL( U93_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U11_1(x_1) ) = max{0, -2}
POL( U21_1(x_1) ) = 2x_1 + 2
POL( U31_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, 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:

   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)


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

(211)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

(213)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(214) 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) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = 2x_2 + 2
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( 0 ) = 0
POL( tt ) = 2
POL( U42_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U52_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U91_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2
POL( U92_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4 + 2
POL( U93_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U11_1(x_1) ) = max{0, -2}
POL( U21_1(x_1) ) = 2
POL( U31_1(x_1) ) = 0
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(215)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(216) 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(isNatList(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( U41_2(x_1, x_2) ) = 2
POL( U71_3(x_1, ..., x_3) ) = 2
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 1}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 1}
POL( 0 ) = 0
POL( tt ) = 2
POL( U42_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( U51_2(x_1, x_2) ) = 2x_1 + x_2 + 1
POL( U52_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U62_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, -2}
POL( U91_4(x_1, ..., x_4) ) = max{0, x_2 + x_4 - 2}
POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U93_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U11_1(x_1) ) = 2
POL( U21_1(x_1) ) = max{0, 2x_1 - 2}
POL( U31_1(x_1) ) = 2
POL( U81_1(x_1) ) = x_1 + 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

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

   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)


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

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

   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U81(tt)) -> mark(nil)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(218) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.


Strictly oriented rules of the TRS R:

   active(U81(tt)) -> mark(nil)

Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2*x_1
   POL(U11(x_1)) = 2*x_1
   POL(U21(x_1)) = x_1
   POL(U31(x_1)) = 2*x_1
   POL(U41(x_1, x_2)) = 2*x_1
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = 2*x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1, x_2, x_3)) = 2*x_1 + x_2
   POL(U72(x_1, x_2)) = 2*x_1 + x_2
   POL(U81(x_1)) = 1 + 2*x_1
   POL(U91(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + 2*x_3 + x_4
   POL(U92(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + 2*x_3 + x_4
   POL(U93(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + x_2 + 2*x_3 + x_4
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(tt) = 0
   POL(zeros) = 0


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

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

   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(220) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.


Strictly oriented rules of the TRS R:

   active(take(0, IL)) -> mark(U81(isNatIList(IL)))

Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2*x_1
   POL(U11(x_1)) = x_1
   POL(U21(x_1)) = 2*x_1
   POL(U31(x_1)) = 2*x_1
   POL(U41(x_1, x_2)) = x_1
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = 2*x_1
   POL(U61(x_1, x_2)) = 2*x_1
   POL(U62(x_1)) = 2*x_1
   POL(U71(x_1, x_2, x_3)) = 2*x_1 + 2*x_2
   POL(U72(x_1, x_2)) = x_1 + 2*x_2
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + x_4
   POL(U92(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + x_4
   POL(U93(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + x_4
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 2*x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 2 + x_1 + x_2
   POL(tt) = 0
   POL(zeros) = 0


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

(221)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(222) 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( U41_2(x_1, x_2) ) = max{0, -2}
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U51_2(x_1, x_2) ) = x_2 + 2
POL( U52_1(x_1) ) = x_1 + 2
POL( isNatList_1(x_1) ) = 0
POL( U61_2(x_1, x_2) ) = x_1 + x_2
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U91_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_4 - 2}
POL( U92_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_4 - 2}
POL( U93_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( take_2(x_1, x_2) ) = 0
POL( U11_1(x_1) ) = 2
POL( U21_1(x_1) ) = 0
POL( U31_1(x_1) ) = max{0, x_1 - 2}
POL( U81_1(x_1) ) = x_1 + 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(223)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(224) 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(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U41_2(x_1, x_2) ) = x_2 + 2
POL( U71_3(x_1, ..., x_3) ) = 1
POL( U72_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2x_1
POL( zeros ) = 0
POL( active_1(x_1) ) = 2x_1
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( U51_2(x_1, x_2) ) = x_1 + 2
POL( U52_1(x_1) ) = max{0, 2x_1 - 1}
POL( isNatList_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U91_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + x_4 - 2}
POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U93_4(x_1, ..., x_4) ) = max{0, x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U11_1(x_1) ) = x_1 + 2
POL( U21_1(x_1) ) = x_1 + 1
POL( U31_1(x_1) ) = max{0, x_1 - 1}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

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

   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)


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

(225)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U42(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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

(227)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U41(X1, X2)) -> ACTIVE(U41(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( U41_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U71_3(x_1, ..., x_3) ) = 1
POL( U72_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 1
POL( zeros ) = 0
POL( active_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( tt ) = 0
POL( U42_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U52_1(x_1) ) = 2x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U61_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1 + 1
POL( U91_4(x_1, ..., x_4) ) = x_1 + 2
POL( U92_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( U93_4(x_1, ..., x_4) ) = x_3 + 2
POL( take_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U11_1(x_1) ) = 1
POL( U21_1(x_1) ) = 2x_1 + 1
POL( U31_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(229)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2)))
   MARK(U42(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U41(tt, V2)) -> MARK(U42(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, 2x_1 - 2}
POL( U71_3(x_1, ..., x_3) ) = max{0, -2}
POL( U72_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 1}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( 0 ) = 2
POL( U41_2(x_1, x_2) ) = 2x_1 + 2
POL( tt ) = 2
POL( U42_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = x_1
POL( U61_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( U62_1(x_1) ) = x_1 + 2
POL( isNat_1(x_1) ) = 2x_1 + 2
POL( s_1(x_1) ) = max{0, -2}
POL( U91_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( U92_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_4 + 2
POL( U93_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = 2x_2 + 2
POL( U11_1(x_1) ) = x_1 + 2
POL( U21_1(x_1) ) = x_1 + 2
POL( U31_1(x_1) ) = max{0, x_1 - 2}
POL( U81_1(x_1) ) = x_1 + 2
POL( nil ) = 2
POL( MARK_1(x_1) ) = max{0, -2}

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

   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)


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

(231)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U42(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U42(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
MARK(x1)  =  x1

U42(x1)  =  U42(x1)

U71(x1, x2, x3)  =  U71

ACTIVE(x1)  =  ACTIVE

mark(x1)  =  mark

tt  =  tt

U72(x1, x2)  =  U72

isNat(x1)  =  isNat(x1)

s(x1)  =  x1

length(x1)  =  length

cons(x1, x2)  =  cons

isNatList(x1)  =  isNatList

zeros  =  zeros

active(x1)  =  active

0  =  0

U41(x1, x2)  =  U41(x1, x2)

isNatIList(x1)  =  x1

U51(x1, x2)  =  U51(x1, x2)

U52(x1)  =  U52(x1)

U61(x1, x2)  =  x2

U62(x1)  =  U62(x1)

U91(x1, x2, x3, x4)  =  U91(x2, x3)

U92(x1, x2, x3, x4)  =  U92(x1, x2, x3, x4)

U93(x1, x2, x3, x4)  =  x2

take(x1, x2)  =  x2

U11(x1)  =  U11(x1)

U21(x1)  =  U21(x1)

U31(x1)  =  U31

U81(x1)  =  U81(x1)

nil  =  nil


Recursive path order with status [RPO].
Quasi-Precedence: [isNatList, zeros, active] > [U71, ACTIVE, U72, length] > [mark, U41_2, U31, U81_1] > [isNat_1, cons, U92_4, U21_1] > U51_2 > U11_1
[isNatList, zeros, active] > 0 > [mark, U41_2, U31, U81_1] > [isNat_1, cons, U92_4, U21_1] > U51_2 > U11_1
[isNatList, zeros, active] > U52_1 > [U42_1, tt] > [mark, U41_2, U31, U81_1] > [isNat_1, cons, U92_4, U21_1] > U51_2 > U11_1
[isNatList, zeros, active] > U62_1 > [U42_1, tt] > [mark, U41_2, U31, U81_1] > [isNat_1, cons, U92_4, U21_1] > U51_2 > U11_1
[isNatList, zeros, active] > U91_2 > [mark, U41_2, U31, U81_1] > [isNat_1, cons, U92_4, U21_1] > U51_2 > U11_1
nil > [mark, U41_2, U31, U81_1] > [isNat_1, cons, U92_4, U21_1] > U51_2 > U11_1

Status: U42_1: multiset status
U71: multiset status
ACTIVE: multiset status
mark: []
tt: multiset status
U72: multiset status
isNat_1: multiset status
length: multiset status
cons: multiset status
isNatList: multiset status
zeros: multiset status
active: []
0: multiset status
U41_2: multiset status
U51_2: multiset status
U52_1: multiset status
U62_1: multiset status
U91_2: multiset status
U92_4: multiset status
U11_1: multiset status
U21_1: multiset status
U31: multiset status
U81_1: multiset status
nil: multiset status


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(233)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(s(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U71(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(tt) =  	[[2A]]
>>>

   <<<
 POL(U72(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1
>>>

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

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

   <<<
 POL(isNatList(x_1)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(zeros) =  	[[0A]]
>>>

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

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U91(x_1, x_2, x_3, x_4)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[0A]] 	* 	x_4
>>>

   <<<
 POL(U92(x_1, x_2, x_3, x_4)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[0A]] 	* 	x_4
>>>

   <<<
 POL(U93(x_1, x_2, x_3, x_4)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[0A]] 	* 	x_4
>>>

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

   <<<
 POL(U11(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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


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

   mark(zeros) -> active(zeros)
   active(zeros) -> mark(cons(0, zeros))
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   U42(mark(X)) -> U42(X)
   U42(active(X)) -> U42(X)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(mark(X)) -> U11(X)
   U11(active(X)) -> U11(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X)) -> U31(X)
   U31(active(X)) -> U31(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


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

(235)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(U72(tt, L)) -> MARK(s(length(L)))
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(236) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(237)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(238) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) we obtained the following new rules [LPAR04]:

   (MARK(U71(isNatList(z1), z1, z0)) -> ACTIVE(U71(mark(isNatList(z1)), z1, z0)),MARK(U71(isNatList(z1), z1, z0)) -> ACTIVE(U71(mark(isNatList(z1)), z1, z0)))


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

(239)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
   MARK(U71(isNatList(z1), z1, z0)) -> ACTIVE(U71(mark(isNatList(z1)), z1, z0))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L))
   ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  x1

U71(x1, x2, x3)  =  U71

MARK(x1)  =  x1

U72(x1, x2)  =  U72

length(x1)  =  length


Knuth-Bendix order [KBO] with precedence:trivial

and weight map:

   length=3
   U71=2
   U72=1

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

   U72(mark(X1), X2) -> U72(X1, X2)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)


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

(241)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2))
   MARK(U71(isNatList(z1), z1, z0)) -> ACTIVE(U71(mark(isNatList(z1)), z1, z0))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   mark(U42(X)) -> active(U42(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
   mark(U72(X1, X2)) -> active(U72(mark(X1), X2))
   active(U72(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
   mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4))
   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
   mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4))
   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   mark(U11(X)) -> active(U11(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   mark(U31(X)) -> active(U31(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U81(X)) -> active(U81(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
   mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U11(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U42(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U42(active(X)) -> U42(X)
   U42(mark(X)) -> U42(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U72(X1, mark(X2)) -> U72(X1, X2)
   U72(mark(X1), X2) -> U72(X1, X2)
   U72(active(X1), X2) -> U72(X1, X2)
   U72(X1, active(X2)) -> U72(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4)
   U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4)
   U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4)
   U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4)
   U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4)
   U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4)
   U11(active(X)) -> U11(X)
   U11(mark(X)) -> U11(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(active(X)) -> U31(X)
   U31(mark(X)) -> U31(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4)
   U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4)
   U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U11(tt))
   active(U21(tt))
   active(U31(tt))
   active(U41(tt, x0))
   active(U42(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0))
   active(U81(tt))
   active(U91(tt, x0, x1, x2))
   active(U92(tt, x0, x1, x2))
   active(U93(tt, x0, x1, x2))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U11(x0))
   mark(tt)
   mark(U21(x0))
   mark(U31(x0))
   mark(U41(x0, x1))
   mark(U42(x0))
   mark(isNatIList(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(isNatList(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1))
   mark(isNat(x0))
   mark(s(x0))
   mark(length(x0))
   mark(U81(x0))
   mark(nil)
   mark(U91(x0, x1, x2, x3))
   mark(U92(x0, x1, x2, x3))
   mark(U93(x0, x1, x2, x3))
   mark(take(x0, x1))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U11(mark(x0))
   U11(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0))
   U31(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U42(mark(x0))
   U42(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1)
   U72(x0, mark(x1))
   U72(active(x0), x1)
   U72(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2, x3)
   U91(x0, mark(x1), x2, x3)
   U91(x0, x1, mark(x2), x3)
   U91(x0, x1, x2, mark(x3))
   U91(active(x0), x1, x2, x3)
   U91(x0, active(x1), x2, x3)
   U91(x0, x1, active(x2), x3)
   U91(x0, x1, x2, active(x3))
   U92(mark(x0), x1, x2, x3)
   U92(x0, mark(x1), x2, x3)
   U92(x0, x1, mark(x2), x3)
   U92(x0, x1, x2, mark(x3))
   U92(active(x0), x1, x2, x3)
   U92(x0, active(x1), x2, x3)
   U92(x0, x1, active(x2), x3)
   U92(x0, x1, x2, active(x3))
   U93(mark(x0), x1, x2, x3)
   U93(x0, mark(x1), x2, x3)
   U93(x0, x1, mark(x2), x3)
   U93(x0, x1, x2, mark(x3))
   U93(active(x0), x1, x2, x3)
   U93(x0, active(x1), x2, x3)
   U93(x0, x1, active(x2), x3)
   U93(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))

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

(242) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(243)
TRUE