NO
proof of Transformed_CSR_04_OvConsOS_complete-noand_Z.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 2787 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPOrderProof [EQUIVALENT, 658 ms]
        (11) QDP
        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
        (13) TRUE
    (14) QDP
        (15) QDPOrderProof [EQUIVALENT, 2631 ms]
        (16) QDP
        (17) DependencyGraphProof [EQUIVALENT, 0 ms]
        (18) AND
            (19) QDP
                (20) QDPOrderProof [EQUIVALENT, 666 ms]
                (21) QDP
                (22) PisEmptyProof [EQUIVALENT, 0 ms]
                (23) YES
            (24) QDP
                (25) QDPOrderProof [EQUIVALENT, 839 ms]
                (26) QDP
                (27) DependencyGraphProof [EQUIVALENT, 0 ms]
                (28) TRUE
            (29) QDP
                (30) TransformationProof [EQUIVALENT, 0 ms]
                (31) QDP
                (32) TransformationProof [EQUIVALENT, 0 ms]
                (33) QDP
                (34) TransformationProof [EQUIVALENT, 0 ms]
                (35) QDP
                (36) DependencyGraphProof [EQUIVALENT, 0 ms]
                (37) QDP
                (38) TransformationProof [EQUIVALENT, 0 ms]
                (39) QDP
                (40) DependencyGraphProof [EQUIVALENT, 0 ms]
                (41) QDP
                (42) TransformationProof [EQUIVALENT, 0 ms]
                (43) QDP
                (44) TransformationProof [EQUIVALENT, 0 ms]
                (45) QDP
                (46) DependencyGraphProof [EQUIVALENT, 0 ms]
                (47) QDP
                (48) TransformationProof [EQUIVALENT, 0 ms]
                (49) QDP
                (50) DependencyGraphProof [EQUIVALENT, 0 ms]
                (51) QDP
                (52) TransformationProof [EQUIVALENT, 0 ms]
                (53) QDP
                (54) TransformationProof [EQUIVALENT, 0 ms]
                (55) QDP
                (56) TransformationProof [EQUIVALENT, 0 ms]
                (57) QDP
                (58) DependencyGraphProof [EQUIVALENT, 0 ms]
                (59) QDP
                (60) TransformationProof [EQUIVALENT, 0 ms]
                (61) QDP
                (62) DependencyGraphProof [EQUIVALENT, 0 ms]
                (63) QDP
                (64) TransformationProof [EQUIVALENT, 0 ms]
                (65) QDP
                (66) DependencyGraphProof [EQUIVALENT, 0 ms]
                (67) QDP
                (68) TransformationProof [EQUIVALENT, 0 ms]
                (69) QDP
                (70) TransformationProof [EQUIVALENT, 0 ms]
                (71) QDP
                (72) DependencyGraphProof [EQUIVALENT, 0 ms]
                (73) QDP
                (74) TransformationProof [EQUIVALENT, 0 ms]
                (75) QDP
                (76) DependencyGraphProof [EQUIVALENT, 0 ms]
                (77) QDP
                (78) TransformationProof [EQUIVALENT, 0 ms]
                (79) QDP
                (80) QDPOrderProof [EQUIVALENT, 469 ms]
                (81) QDP
                (82) QDPOrderProof [EQUIVALENT, 346 ms]
                (83) QDP
                (84) QDPOrderProof [EQUIVALENT, 496 ms]
                (85) QDP
                (86) QDPOrderProof [EQUIVALENT, 1005 ms]
                (87) QDP
                (88) NonTerminationLoopProof [COMPLETE, 162 ms]
                (89) NO
            (90) QDP
                (91) TransformationProof [EQUIVALENT, 0 ms]
                (92) QDP
                (93) TransformationProof [EQUIVALENT, 0 ms]
                (94) QDP
                (95) TransformationProof [EQUIVALENT, 0 ms]
                (96) QDP
                (97) DependencyGraphProof [EQUIVALENT, 0 ms]
                (98) QDP
                (99) TransformationProof [EQUIVALENT, 0 ms]
                (100) QDP
                (101) DependencyGraphProof [EQUIVALENT, 0 ms]
                (102) QDP
                (103) TransformationProof [EQUIVALENT, 0 ms]
                (104) QDP
                (105) DependencyGraphProof [EQUIVALENT, 0 ms]
                (106) QDP
                (107) TransformationProof [EQUIVALENT, 0 ms]
                (108) QDP
                (109) DependencyGraphProof [EQUIVALENT, 0 ms]
                (110) QDP
                (111) TransformationProof [EQUIVALENT, 0 ms]
                (112) QDP
                (113) TransformationProof [EQUIVALENT, 0 ms]
                (114) QDP
                (115) DependencyGraphProof [EQUIVALENT, 0 ms]
                (116) QDP
                (117) TransformationProof [EQUIVALENT, 0 ms]
                (118) QDP
                (119) TransformationProof [EQUIVALENT, 0 ms]
                (120) QDP
                (121) TransformationProof [EQUIVALENT, 0 ms]
                (122) QDP
                (123) TransformationProof [EQUIVALENT, 0 ms]
                (124) QDP
                (125) DependencyGraphProof [EQUIVALENT, 0 ms]
                (126) QDP
                (127) TransformationProof [EQUIVALENT, 0 ms]
                (128) QDP
                (129) TransformationProof [EQUIVALENT, 0 ms]
                (130) QDP
                (131) DependencyGraphProof [EQUIVALENT, 0 ms]
                (132) QDP
                (133) TransformationProof [EQUIVALENT, 0 ms]
                (134) QDP
                (135) DependencyGraphProof [EQUIVALENT, 0 ms]
                (136) QDP
                (137) TransformationProof [EQUIVALENT, 0 ms]
                (138) QDP
                (139) DependencyGraphProof [EQUIVALENT, 0 ms]
                (140) QDP
                (141) QDPOrderProof [EQUIVALENT, 648 ms]
                (142) QDP
                (143) QDPOrderProof [EQUIVALENT, 511 ms]
                (144) QDP
                (145) QDPOrderProof [EQUIVALENT, 830 ms]
                (146) QDP
                (147) QDPOrderProof [EQUIVALENT, 1179 ms]
                (148) QDP
            (149) QDP
                (150) TransformationProof [EQUIVALENT, 0 ms]
                (151) QDP
                (152) TransformationProof [EQUIVALENT, 0 ms]
                (153) QDP
                (154) TransformationProof [EQUIVALENT, 0 ms]
                (155) QDP
                (156) DependencyGraphProof [EQUIVALENT, 0 ms]
                (157) QDP
                (158) TransformationProof [EQUIVALENT, 0 ms]
                (159) QDP
                (160) DependencyGraphProof [EQUIVALENT, 0 ms]
                (161) QDP
                (162) TransformationProof [EQUIVALENT, 0 ms]
                (163) QDP
                (164) DependencyGraphProof [EQUIVALENT, 0 ms]
                (165) QDP
                (166) TransformationProof [EQUIVALENT, 0 ms]
                (167) QDP
                (168) DependencyGraphProof [EQUIVALENT, 0 ms]
                (169) QDP
                (170) TransformationProof [EQUIVALENT, 0 ms]
                (171) QDP
                (172) TransformationProof [EQUIVALENT, 0 ms]
                (173) QDP
                (174) TransformationProof [EQUIVALENT, 0 ms]
                (175) QDP
                (176) DependencyGraphProof [EQUIVALENT, 0 ms]
                (177) QDP
                (178) TransformationProof [EQUIVALENT, 0 ms]
                (179) QDP
                (180) TransformationProof [EQUIVALENT, 0 ms]
                (181) QDP
                (182) TransformationProof [EQUIVALENT, 0 ms]
                (183) QDP
                (184) DependencyGraphProof [EQUIVALENT, 0 ms]
                (185) QDP
                (186) TransformationProof [EQUIVALENT, 0 ms]
                (187) QDP
                (188) TransformationProof [EQUIVALENT, 0 ms]
                (189) QDP
                (190) DependencyGraphProof [EQUIVALENT, 0 ms]
                (191) QDP
                (192) TransformationProof [EQUIVALENT, 0 ms]
                (193) QDP
                (194) DependencyGraphProof [EQUIVALENT, 0 ms]
                (195) QDP
                (196) TransformationProof [EQUIVALENT, 0 ms]
                (197) QDP
                (198) DependencyGraphProof [EQUIVALENT, 0 ms]
                (199) QDP
                (200) QDPOrderProof [EQUIVALENT, 660 ms]
                (201) QDP
                (202) QDPOrderProof [EQUIVALENT, 695 ms]
                (203) QDP
                (204) QDPOrderProof [EQUIVALENT, 625 ms]
                (205) QDP
                (206) QDPOrderProof [EQUIVALENT, 598 ms]
                (207) QDP


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

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

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

Q is empty.

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

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

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

   ZEROS -> CONS(0, n__zeros)
   ZEROS -> 0^1
   U101^1(tt, V1, V2) -> U102^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U101^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ACTIVATE(V2)
   U102^1(tt, V1, V2) -> U103^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U102^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U102^1(tt, V1, V2) -> ACTIVATE(V2)
   U102^1(tt, V1, V2) -> ACTIVATE(V1)
   U103^1(tt, V1, V2) -> U104^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U103^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ACTIVATE(V1)
   U104^1(tt, V1, V2) -> U105^1(isNat(activate(V1)), activate(V2))
   U104^1(tt, V1, V2) -> ISNAT(activate(V1))
   U104^1(tt, V1, V2) -> ACTIVATE(V1)
   U104^1(tt, V1, V2) -> ACTIVATE(V2)
   U105^1(tt, V2) -> U106^1(isNatIList(activate(V2)))
   U105^1(tt, V2) -> ISNATILIST(activate(V2))
   U105^1(tt, V2) -> ACTIVATE(V2)
   U11^1(tt, V1) -> U12^1(isNatIListKind(activate(V1)), activate(V1))
   U11^1(tt, V1) -> ISNATILISTKIND(activate(V1))
   U11^1(tt, V1) -> ACTIVATE(V1)
   U111^1(tt, L, N) -> U112^1(isNatIListKind(activate(L)), activate(L), activate(N))
   U111^1(tt, L, N) -> ISNATILISTKIND(activate(L))
   U111^1(tt, L, N) -> ACTIVATE(L)
   U111^1(tt, L, N) -> ACTIVATE(N)
   U112^1(tt, L, N) -> U113^1(isNat(activate(N)), activate(L), activate(N))
   U112^1(tt, L, N) -> ISNAT(activate(N))
   U112^1(tt, L, N) -> ACTIVATE(N)
   U112^1(tt, L, N) -> ACTIVATE(L)
   U113^1(tt, L, N) -> U114^1(isNatKind(activate(N)), activate(L))
   U113^1(tt, L, N) -> ISNATKIND(activate(N))
   U113^1(tt, L, N) -> ACTIVATE(N)
   U113^1(tt, L, N) -> ACTIVATE(L)
   U114^1(tt, L) -> S(length(activate(L)))
   U114^1(tt, L) -> LENGTH(activate(L))
   U114^1(tt, L) -> ACTIVATE(L)
   U12^1(tt, V1) -> U13^1(isNatList(activate(V1)))
   U12^1(tt, V1) -> ISNATLIST(activate(V1))
   U12^1(tt, V1) -> ACTIVATE(V1)
   U121^1(tt, IL) -> U122^1(isNatIListKind(activate(IL)))
   U121^1(tt, IL) -> ISNATILISTKIND(activate(IL))
   U121^1(tt, IL) -> ACTIVATE(IL)
   U122^1(tt) -> NIL
   U131^1(tt, IL, M, N) -> U132^1(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL))
   U131^1(tt, IL, M, N) -> ACTIVATE(IL)
   U131^1(tt, IL, M, N) -> ACTIVATE(M)
   U131^1(tt, IL, M, N) -> ACTIVATE(N)
   U132^1(tt, IL, M, N) -> U133^1(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U132^1(tt, IL, M, N) -> ISNAT(activate(M))
   U132^1(tt, IL, M, N) -> ACTIVATE(M)
   U132^1(tt, IL, M, N) -> ACTIVATE(IL)
   U132^1(tt, IL, M, N) -> ACTIVATE(N)
   U133^1(tt, IL, M, N) -> U134^1(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U133^1(tt, IL, M, N) -> ISNATKIND(activate(M))
   U133^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ACTIVATE(IL)
   U133^1(tt, IL, M, N) -> ACTIVATE(N)
   U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U134^1(tt, IL, M, N) -> ISNAT(activate(N))
   U134^1(tt, IL, M, N) -> ACTIVATE(N)
   U134^1(tt, IL, M, N) -> ACTIVATE(IL)
   U134^1(tt, IL, M, N) -> ACTIVATE(M)
   U135^1(tt, IL, M, N) -> U136^1(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U135^1(tt, IL, M, N) -> ISNATKIND(activate(N))
   U135^1(tt, IL, M, N) -> ACTIVATE(N)
   U135^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ACTIVATE(M)
   U136^1(tt, IL, M, N) -> CONS(activate(N), n__take(activate(M), activate(IL)))
   U136^1(tt, IL, M, N) -> ACTIVATE(N)
   U136^1(tt, IL, M, N) -> ACTIVATE(M)
   U136^1(tt, IL, M, N) -> ACTIVATE(IL)
   U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1))
   U21^1(tt, V1) -> ISNATKIND(activate(V1))
   U21^1(tt, V1) -> ACTIVATE(V1)
   U22^1(tt, V1) -> U23^1(isNat(activate(V1)))
   U22^1(tt, V1) -> ISNAT(activate(V1))
   U22^1(tt, V1) -> ACTIVATE(V1)
   U31^1(tt, V) -> U32^1(isNatIListKind(activate(V)), activate(V))
   U31^1(tt, V) -> ISNATILISTKIND(activate(V))
   U31^1(tt, V) -> ACTIVATE(V)
   U32^1(tt, V) -> U33^1(isNatList(activate(V)))
   U32^1(tt, V) -> ISNATLIST(activate(V))
   U32^1(tt, V) -> ACTIVATE(V)
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U41^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ACTIVATE(V2)
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U42^1(tt, V1, V2) -> ACTIVATE(V2)
   U42^1(tt, V1, V2) -> ACTIVATE(V1)
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U43^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2))
   U44^1(tt, V1, V2) -> ISNAT(activate(V1))
   U44^1(tt, V1, V2) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V2)
   U45^1(tt, V2) -> U46^1(isNatIList(activate(V2)))
   U45^1(tt, V2) -> ISNATILIST(activate(V2))
   U45^1(tt, V2) -> ACTIVATE(V2)
   U51^1(tt, V2) -> U52^1(isNatIListKind(activate(V2)))
   U51^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   U51^1(tt, V2) -> ACTIVATE(V2)
   U61^1(tt, V2) -> U62^1(isNatIListKind(activate(V2)))
   U61^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   U61^1(tt, V2) -> ACTIVATE(V2)
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U91^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ACTIVATE(V2)
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U92^1(tt, V1, V2) -> ACTIVATE(V2)
   U92^1(tt, V1, V2) -> ACTIVATE(V1)
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U93^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ACTIVATE(V1)
   U94^1(tt, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2))
   U94^1(tt, V1, V2) -> ISNAT(activate(V1))
   U94^1(tt, V1, V2) -> ACTIVATE(V1)
   U94^1(tt, V1, V2) -> ACTIVATE(V2)
   U95^1(tt, V2) -> U96^1(isNatList(activate(V2)))
   U95^1(tt, V2) -> ISNATLIST(activate(V2))
   U95^1(tt, V2) -> ACTIVATE(V2)
   ISNAT(n__length(V1)) -> U11^1(isNatIListKind(activate(V1)), activate(V1))
   ISNAT(n__length(V1)) -> ISNATILISTKIND(activate(V1))
   ISNAT(n__length(V1)) -> ACTIVATE(V1)
   ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1))
   ISNAT(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNAT(n__s(V1)) -> ACTIVATE(V1)
   ISNATILIST(V) -> U31^1(isNatIListKind(activate(V)), activate(V))
   ISNATILIST(V) -> ISNATILISTKIND(activate(V))
   ISNATILIST(V) -> ACTIVATE(V)
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   ISNATILIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2))
   ISNATILISTKIND(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2))
   ISNATILISTKIND(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V2)
   ISNATKIND(n__length(V1)) -> U71^1(isNatIListKind(activate(V1)))
   ISNATKIND(n__length(V1)) -> ISNATILISTKIND(activate(V1))
   ISNATKIND(n__length(V1)) -> ACTIVATE(V1)
   ISNATKIND(n__s(V1)) -> U81^1(isNatKind(activate(V1)))
   ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATLIST(n__take(V1, V2)) -> U101^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2)
   LENGTH(nil) -> 0^1
   LENGTH(cons(N, L)) -> U111^1(isNatList(activate(L)), activate(L), N)
   LENGTH(cons(N, L)) -> ISNATLIST(activate(L))
   LENGTH(cons(N, L)) -> ACTIVATE(L)
   TAKE(0, IL) -> U121^1(isNatIList(IL), IL)
   TAKE(0, IL) -> ISNATILIST(IL)
   TAKE(s(M), cons(N, IL)) -> U131^1(isNatIList(activate(IL)), activate(IL), M, N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)
   ACTIVATE(n__zeros) -> ZEROS
   ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2)
   ACTIVATE(n__0) -> 0^1
   ACTIVATE(n__length(X)) -> LENGTH(X)
   ACTIVATE(n__s(X)) -> S(X)
   ACTIVATE(n__cons(X1, X2)) -> CONS(X1, X2)
   ACTIVATE(n__nil) -> NIL

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 22 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2)
   TAKE(0, IL) -> U121^1(isNatIList(IL), IL)
   U121^1(tt, IL) -> ISNATILISTKIND(activate(IL))
   ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2))
   U51^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__length(V1)) -> ISNATILISTKIND(activate(V1))
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1)
   ACTIVATE(n__length(X)) -> LENGTH(X)
   LENGTH(cons(N, L)) -> U111^1(isNatList(activate(L)), activate(L), N)
   U111^1(tt, L, N) -> U112^1(isNatIListKind(activate(L)), activate(L), activate(N))
   U112^1(tt, L, N) -> U113^1(isNat(activate(N)), activate(L), activate(N))
   U113^1(tt, L, N) -> U114^1(isNatKind(activate(N)), activate(L))
   U114^1(tt, L) -> LENGTH(activate(L))
   LENGTH(cons(N, L)) -> ISNATLIST(activate(L))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2))
   U95^1(tt, V2) -> ISNATLIST(activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__length(V1)) -> ACTIVATE(V1)
   ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATLIST(n__take(V1, V2)) -> U101^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101^1(tt, V1, V2) -> U102^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102^1(tt, V1, V2) -> U103^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103^1(tt, V1, V2) -> U104^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104^1(tt, V1, V2) -> U105^1(isNat(activate(V1)), activate(V2))
   U105^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(V) -> U31^1(isNatIListKind(activate(V)), activate(V))
   U31^1(tt, V) -> U32^1(isNatIListKind(activate(V)), activate(V))
   U32^1(tt, V) -> ISNATLIST(activate(V))
   ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2)
   U32^1(tt, V) -> ACTIVATE(V)
   U31^1(tt, V) -> ISNATILISTKIND(activate(V))
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2))
   U61^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V2)
   U61^1(tt, V2) -> ACTIVATE(V2)
   U31^1(tt, V) -> ACTIVATE(V)
   ISNATILIST(V) -> ISNATILISTKIND(activate(V))
   ISNATILIST(V) -> ACTIVATE(V)
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2))
   U45^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   U45^1(tt, V2) -> ACTIVATE(V2)
   U44^1(tt, V1, V2) -> ISNAT(activate(V1))
   ISNAT(n__length(V1)) -> U11^1(isNatIListKind(activate(V1)), activate(V1))
   U11^1(tt, V1) -> U12^1(isNatIListKind(activate(V1)), activate(V1))
   U12^1(tt, V1) -> ISNATLIST(activate(V1))
   U12^1(tt, V1) -> ACTIVATE(V1)
   U11^1(tt, V1) -> ISNATILISTKIND(activate(V1))
   U11^1(tt, V1) -> ACTIVATE(V1)
   ISNAT(n__length(V1)) -> ISNATILISTKIND(activate(V1))
   ISNAT(n__length(V1)) -> ACTIVATE(V1)
   ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1))
   U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1))
   U22^1(tt, V1) -> ISNAT(activate(V1))
   ISNAT(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNAT(n__s(V1)) -> ACTIVATE(V1)
   U22^1(tt, V1) -> ACTIVATE(V1)
   U21^1(tt, V1) -> ISNATKIND(activate(V1))
   U21^1(tt, V1) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U43^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ACTIVATE(V1)
   U42^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U42^1(tt, V1, V2) -> ACTIVATE(V2)
   U42^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U41^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ACTIVATE(V2)
   U105^1(tt, V2) -> ACTIVATE(V2)
   U104^1(tt, V1, V2) -> ISNAT(activate(V1))
   U104^1(tt, V1, V2) -> ACTIVATE(V1)
   U104^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U103^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ACTIVATE(V1)
   U102^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U102^1(tt, V1, V2) -> ACTIVATE(V2)
   U102^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U101^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ACTIVATE(V2)
   U95^1(tt, V2) -> ACTIVATE(V2)
   U94^1(tt, V1, V2) -> ISNAT(activate(V1))
   U94^1(tt, V1, V2) -> ACTIVATE(V1)
   U94^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U93^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ACTIVATE(V1)
   U92^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U92^1(tt, V1, V2) -> ACTIVATE(V2)
   U92^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U91^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ACTIVATE(V2)
   LENGTH(cons(N, L)) -> ACTIVATE(L)
   U114^1(tt, L) -> ACTIVATE(L)
   U113^1(tt, L, N) -> ISNATKIND(activate(N))
   U113^1(tt, L, N) -> ACTIVATE(N)
   U113^1(tt, L, N) -> ACTIVATE(L)
   U112^1(tt, L, N) -> ISNAT(activate(N))
   U112^1(tt, L, N) -> ACTIVATE(N)
   U112^1(tt, L, N) -> ACTIVATE(L)
   U111^1(tt, L, N) -> ISNATILISTKIND(activate(L))
   U111^1(tt, L, N) -> ACTIVATE(L)
   U111^1(tt, L, N) -> ACTIVATE(N)
   U51^1(tt, V2) -> ACTIVATE(V2)
   U121^1(tt, IL) -> ACTIVATE(IL)
   TAKE(0, IL) -> ISNATILIST(IL)
   TAKE(s(M), cons(N, IL)) -> U131^1(isNatIList(activate(IL)), activate(IL), M, N)
   U131^1(tt, IL, M, N) -> U132^1(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132^1(tt, IL, M, N) -> U133^1(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133^1(tt, IL, M, N) -> U134^1(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135^1(tt, IL, M, N) -> U136^1(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136^1(tt, IL, M, N) -> ACTIVATE(N)
   U136^1(tt, IL, M, N) -> ACTIVATE(M)
   U136^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ISNATKIND(activate(N))
   U135^1(tt, IL, M, N) -> ACTIVATE(N)
   U135^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ACTIVATE(M)
   U134^1(tt, IL, M, N) -> ISNAT(activate(N))
   U134^1(tt, IL, M, N) -> ACTIVATE(N)
   U134^1(tt, IL, M, N) -> ACTIVATE(IL)
   U134^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ISNATKIND(activate(M))
   U133^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ACTIVATE(IL)
   U133^1(tt, IL, M, N) -> ACTIVATE(N)
   U132^1(tt, IL, M, N) -> ISNAT(activate(M))
   U132^1(tt, IL, M, N) -> ACTIVATE(M)
   U132^1(tt, IL, M, N) -> ACTIVATE(IL)
   U132^1(tt, IL, M, N) -> ACTIVATE(N)
   U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL))
   U131^1(tt, IL, M, N) -> ACTIVATE(IL)
   U131^1(tt, IL, M, N) -> ACTIVATE(M)
   U131^1(tt, IL, M, N) -> ACTIVATE(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   ISNATKIND(n__length(V1)) -> ISNATILISTKIND(activate(V1))
   LENGTH(cons(N, L)) -> ISNATLIST(activate(L))
   ISNATKIND(n__length(V1)) -> ACTIVATE(V1)
   U11^1(tt, V1) -> U12^1(isNatIListKind(activate(V1)), activate(V1))
   U11^1(tt, V1) -> ISNATILISTKIND(activate(V1))
   U11^1(tt, V1) -> ACTIVATE(V1)
   ISNAT(n__length(V1)) -> ISNATILISTKIND(activate(V1))
   ISNAT(n__length(V1)) -> ACTIVATE(V1)
   LENGTH(cons(N, L)) -> ACTIVATE(L)
   U114^1(tt, L) -> ACTIVATE(L)
   U113^1(tt, L, N) -> ISNATKIND(activate(N))
   U113^1(tt, L, N) -> ACTIVATE(N)
   U113^1(tt, L, N) -> ACTIVATE(L)
   U112^1(tt, L, N) -> ISNAT(activate(N))
   U112^1(tt, L, N) -> ACTIVATE(N)
   U112^1(tt, L, N) -> ACTIVATE(L)
   U111^1(tt, L, N) -> ISNATILISTKIND(activate(L))
   U111^1(tt, L, N) -> ACTIVATE(L)
   U111^1(tt, L, N) -> ACTIVATE(N)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNAT_1(x_1) ) = x_1
POL( ISNATILIST_1(x_1) ) = x_1
POL( ISNATILISTKIND_1(x_1) ) = x_1
POL( ISNATKIND_1(x_1) ) = x_1
POL( ISNATLIST_1(x_1) ) = x_1
POL( LENGTH_1(x_1) ) = x_1 + 2
POL( U101^1_3(x_1, ..., x_3) ) = x_2 + x_3
POL( U102^1_3(x_1, ..., x_3) ) = x_2 + x_3
POL( U103^1_3(x_1, ..., x_3) ) = x_2 + x_3
POL( U104^1_3(x_1, ..., x_3) ) = x_2 + x_3
POL( U105^1_2(x_1, x_2) ) = x_2
POL( U111^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U112^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U113^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U114^1_2(x_1, x_2) ) = x_1 + x_2
POL( U11^1_2(x_1, x_2) ) = x_2 + 2
POL( U121^1_2(x_1, x_2) ) = x_2
POL( U12^1_2(x_1, x_2) ) = x_2
POL( U131^1_4(x_1, ..., x_4) ) = x_2 + x_3 + 2x_4
POL( U132^1_4(x_1, ..., x_4) ) = x_2 + x_3 + 2x_4
POL( U133^1_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + x_3 + 2x_4 - 2}
POL( U134^1_4(x_1, ..., x_4) ) = x_2 + x_3 + 2x_4
POL( U135^1_4(x_1, ..., x_4) ) = x_2 + x_3 + 2x_4
POL( U136^1_4(x_1, ..., x_4) ) = x_2 + x_3 + 2x_4
POL( U21^1_2(x_1, x_2) ) = x_2
POL( U22^1_2(x_1, x_2) ) = x_2
POL( U31^1_2(x_1, x_2) ) = x_2
POL( U32^1_2(x_1, x_2) ) = x_2
POL( U41^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U42^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U43^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U44^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U45^1_2(x_1, x_2) ) = x_2
POL( U51^1_2(x_1, x_2) ) = x_2
POL( U61^1_2(x_1, x_2) ) = x_2
POL( U91^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U92^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U93^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U94^1_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( U95^1_2(x_1, x_2) ) = x_2
POL( isNatIList_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( isNatIListKind_1(x_1) ) = x_1 + 1
POL( activate_1(x_1) ) = x_1
POL( U106_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( n__zeros ) = 2
POL( tt ) = 2
POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = x_1 + x_2
POL( U112_3(x_1, ..., x_3) ) = x_2 + 2
POL( U113_3(x_1, ..., x_3) ) = x_2 + 2
POL( U114_2(x_1, x_2) ) = x_2 + 2
POL( U12_2(x_1, x_2) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U21_2(x_1, x_2) ) = x_1
POL( U22_2(x_1, x_2) ) = x_1
POL( U32_2(x_1, x_2) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = x_2 + 1
POL( U61_2(x_1, x_2) ) = x_2 + 1
POL( U91_3(x_1, ..., x_3) ) = x_1
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( isNat_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = x_1
POL( U71_1(x_1) ) = 2
POL( U96_1(x_1) ) = 2
POL( n__take_2(x_1, x_2) ) = x_1 + 2x_2
POL( length_1(x_1) ) = x_1 + 2
POL( s_1(x_1) ) = x_1
POL( U81_1(x_1) ) = 2
POL( zeros ) = 2
POL( take_2(x_1, x_2) ) = x_1 + 2x_2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__length_1(x_1) ) = x_1 + 2
POL( n__s_1(x_1) ) = x_1
POL( n__nil ) = 1
POL( nil ) = 1
POL( ACTIVATE_1(x_1) ) = x_1
POL( TAKE_2(x_1, x_2) ) = x_1 + x_2

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

   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> U121(isNatIList(IL), IL)
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2)
   TAKE(0, IL) -> U121^1(isNatIList(IL), IL)
   U121^1(tt, IL) -> ISNATILISTKIND(activate(IL))
   ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2))
   U51^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1)
   ACTIVATE(n__length(X)) -> LENGTH(X)
   LENGTH(cons(N, L)) -> U111^1(isNatList(activate(L)), activate(L), N)
   U111^1(tt, L, N) -> U112^1(isNatIListKind(activate(L)), activate(L), activate(N))
   U112^1(tt, L, N) -> U113^1(isNat(activate(N)), activate(L), activate(N))
   U113^1(tt, L, N) -> U114^1(isNatKind(activate(N)), activate(L))
   U114^1(tt, L) -> LENGTH(activate(L))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2))
   U95^1(tt, V2) -> ISNATLIST(activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATLIST(n__take(V1, V2)) -> U101^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101^1(tt, V1, V2) -> U102^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102^1(tt, V1, V2) -> U103^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103^1(tt, V1, V2) -> U104^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104^1(tt, V1, V2) -> U105^1(isNat(activate(V1)), activate(V2))
   U105^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(V) -> U31^1(isNatIListKind(activate(V)), activate(V))
   U31^1(tt, V) -> U32^1(isNatIListKind(activate(V)), activate(V))
   U32^1(tt, V) -> ISNATLIST(activate(V))
   ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2)
   U32^1(tt, V) -> ACTIVATE(V)
   U31^1(tt, V) -> ISNATILISTKIND(activate(V))
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2))
   U61^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V2)
   U61^1(tt, V2) -> ACTIVATE(V2)
   U31^1(tt, V) -> ACTIVATE(V)
   ISNATILIST(V) -> ISNATILISTKIND(activate(V))
   ISNATILIST(V) -> ACTIVATE(V)
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2))
   U45^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   U45^1(tt, V2) -> ACTIVATE(V2)
   U44^1(tt, V1, V2) -> ISNAT(activate(V1))
   ISNAT(n__length(V1)) -> U11^1(isNatIListKind(activate(V1)), activate(V1))
   U12^1(tt, V1) -> ISNATLIST(activate(V1))
   U12^1(tt, V1) -> ACTIVATE(V1)
   ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1))
   U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1))
   U22^1(tt, V1) -> ISNAT(activate(V1))
   ISNAT(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNAT(n__s(V1)) -> ACTIVATE(V1)
   U22^1(tt, V1) -> ACTIVATE(V1)
   U21^1(tt, V1) -> ISNATKIND(activate(V1))
   U21^1(tt, V1) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U43^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ACTIVATE(V1)
   U42^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U42^1(tt, V1, V2) -> ACTIVATE(V2)
   U42^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U41^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ACTIVATE(V2)
   U105^1(tt, V2) -> ACTIVATE(V2)
   U104^1(tt, V1, V2) -> ISNAT(activate(V1))
   U104^1(tt, V1, V2) -> ACTIVATE(V1)
   U104^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U103^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ACTIVATE(V1)
   U102^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U102^1(tt, V1, V2) -> ACTIVATE(V2)
   U102^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U101^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ACTIVATE(V2)
   U95^1(tt, V2) -> ACTIVATE(V2)
   U94^1(tt, V1, V2) -> ISNAT(activate(V1))
   U94^1(tt, V1, V2) -> ACTIVATE(V1)
   U94^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U93^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ACTIVATE(V1)
   U92^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U92^1(tt, V1, V2) -> ACTIVATE(V2)
   U92^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U91^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ACTIVATE(V2)
   U51^1(tt, V2) -> ACTIVATE(V2)
   U121^1(tt, IL) -> ACTIVATE(IL)
   TAKE(0, IL) -> ISNATILIST(IL)
   TAKE(s(M), cons(N, IL)) -> U131^1(isNatIList(activate(IL)), activate(IL), M, N)
   U131^1(tt, IL, M, N) -> U132^1(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132^1(tt, IL, M, N) -> U133^1(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133^1(tt, IL, M, N) -> U134^1(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135^1(tt, IL, M, N) -> U136^1(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136^1(tt, IL, M, N) -> ACTIVATE(N)
   U136^1(tt, IL, M, N) -> ACTIVATE(M)
   U136^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ISNATKIND(activate(N))
   U135^1(tt, IL, M, N) -> ACTIVATE(N)
   U135^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ACTIVATE(M)
   U134^1(tt, IL, M, N) -> ISNAT(activate(N))
   U134^1(tt, IL, M, N) -> ACTIVATE(N)
   U134^1(tt, IL, M, N) -> ACTIVATE(IL)
   U134^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ISNATKIND(activate(M))
   U133^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ACTIVATE(IL)
   U133^1(tt, IL, M, N) -> ACTIVATE(N)
   U132^1(tt, IL, M, N) -> ISNAT(activate(M))
   U132^1(tt, IL, M, N) -> ACTIVATE(M)
   U132^1(tt, IL, M, N) -> ACTIVATE(IL)
   U132^1(tt, IL, M, N) -> ACTIVATE(N)
   U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL))
   U131^1(tt, IL, M, N) -> ACTIVATE(IL)
   U131^1(tt, IL, M, N) -> ACTIVATE(M)
   U131^1(tt, IL, M, N) -> ACTIVATE(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

   U111^1(tt, L, N) -> U112^1(isNatIListKind(activate(L)), activate(L), activate(N))
   U112^1(tt, L, N) -> U113^1(isNat(activate(N)), activate(L), activate(N))
   U113^1(tt, L, N) -> U114^1(isNatKind(activate(N)), activate(L))
   U114^1(tt, L) -> LENGTH(activate(L))
   LENGTH(cons(N, L)) -> U111^1(isNatList(activate(L)), activate(L), N)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   LENGTH(cons(N, L)) -> U111^1(isNatList(activate(L)), activate(L), N)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( LENGTH_1(x_1) ) = 2x_1 + 1
POL( U111^1_3(x_1, ..., x_3) ) = x_1 + 2x_2
POL( U112^1_3(x_1, ..., x_3) ) = x_1 + 2x_2
POL( U113^1_3(x_1, ..., x_3) ) = 2x_2 + 1
POL( U114^1_2(x_1, x_2) ) = 2x_2 + 1
POL( U101_3(x_1, ..., x_3) ) = 2x_2
POL( U102_3(x_1, ..., x_3) ) = 2x_2
POL( U103_3(x_1, ..., x_3) ) = 2x_2
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 1}
POL( U105_2(x_1, x_2) ) = x_1
POL( U11_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( U111_3(x_1, ..., x_3) ) = 2x_2
POL( U112_3(x_1, ..., x_3) ) = 2x_2
POL( U113_3(x_1, ..., x_3) ) = 2x_2
POL( U114_2(x_1, x_2) ) = 2x_2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U131_4(x_1, ..., x_4) ) = 2x_3
POL( U132_4(x_1, ..., x_4) ) = 2x_3
POL( U133_4(x_1, ..., x_4) ) = 2x_3
POL( U134_4(x_1, ..., x_4) ) = 2x_3
POL( U135_4(x_1, ..., x_4) ) = 2x_3
POL( U136_4(x_1, ..., x_4) ) = 2x_3
POL( U21_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( U22_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( U31_2(x_1, x_2) ) = 1
POL( U32_2(x_1, x_2) ) = 1
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2x_1
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( U91_3(x_1, ..., x_3) ) = 2x_3
POL( U92_3(x_1, ..., x_3) ) = 2x_3
POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2x_3
POL( U95_2(x_1, x_2) ) = 2x_2
POL( cons_2(x_1, x_2) ) = 2x_2
POL( isNatList_1(x_1) ) = 2x_1
POL( isNatIListKind_1(x_1) ) = 1
POL( isNat_1(x_1) ) = 2x_1
POL( isNatKind_1(x_1) ) = 1
POL( U106_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 2
POL( U122_1(x_1) ) = max{0, 2x_1 - 1}
POL( U23_1(x_1) ) = x_1
POL( U46_1(x_1) ) = x_1
POL( U52_1(x_1) ) = 1
POL( U62_1(x_1) ) = 1
POL( U71_1(x_1) ) = 1
POL( U81_1(x_1) ) = 1
POL( n__take_2(x_1, x_2) ) = x_1
POL( length_1(x_1) ) = x_1
POL( s_1(x_1) ) = 2x_1
POL( U13_1(x_1) ) = x_1
POL( U33_1(x_1) ) = 1
POL( U96_1(x_1) ) = x_1
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( take_2(x_1, x_2) ) = x_1
POL( n__0 ) = 1
POL( 0 ) = 1
POL( n__length_1(x_1) ) = x_1
POL( n__s_1(x_1) ) = 2x_1
POL( n__cons_2(x_1, x_2) ) = 2x_2
POL( n__nil ) = 1
POL( nil ) = 1
POL( tt ) = 1
POL( U121_2(x_1, x_2) ) = 1

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> U121(isNatIList(IL), IL)
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U111^1(tt, L, N) -> U112^1(isNatIListKind(activate(L)), activate(L), activate(N))
   U112^1(tt, L, N) -> U113^1(isNat(activate(N)), activate(L), activate(N))
   U113^1(tt, L, N) -> U114^1(isNatKind(activate(N)), activate(L))
   U114^1(tt, L) -> LENGTH(activate(L))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(12) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
----------------------------------------

(13)
TRUE

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

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

   TAKE(0, IL) -> U121^1(isNatIList(IL), IL)
   U121^1(tt, IL) -> ISNATILISTKIND(activate(IL))
   ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2))
   U51^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ACTIVATE(V1)
   ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2)
   TAKE(0, IL) -> ISNATILIST(IL)
   ISNATILIST(V) -> U31^1(isNatIListKind(activate(V)), activate(V))
   U31^1(tt, V) -> U32^1(isNatIListKind(activate(V)), activate(V))
   U32^1(tt, V) -> ISNATLIST(activate(V))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2))
   U95^1(tt, V2) -> ISNATLIST(activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATLIST(n__take(V1, V2)) -> U101^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101^1(tt, V1, V2) -> U102^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102^1(tt, V1, V2) -> U103^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103^1(tt, V1, V2) -> U104^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104^1(tt, V1, V2) -> U105^1(isNat(activate(V1)), activate(V2))
   U105^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(V) -> ISNATILISTKIND(activate(V))
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2))
   U61^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V2)
   U61^1(tt, V2) -> ACTIVATE(V2)
   ISNATILIST(V) -> ACTIVATE(V)
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2))
   U45^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   U45^1(tt, V2) -> ACTIVATE(V2)
   U44^1(tt, V1, V2) -> ISNAT(activate(V1))
   ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1))
   U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1))
   U22^1(tt, V1) -> ISNAT(activate(V1))
   ISNAT(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNAT(n__s(V1)) -> ACTIVATE(V1)
   U22^1(tt, V1) -> ACTIVATE(V1)
   U21^1(tt, V1) -> ISNATKIND(activate(V1))
   U21^1(tt, V1) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U43^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ACTIVATE(V1)
   U42^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U42^1(tt, V1, V2) -> ACTIVATE(V2)
   U42^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U41^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ACTIVATE(V2)
   U105^1(tt, V2) -> ACTIVATE(V2)
   U104^1(tt, V1, V2) -> ISNAT(activate(V1))
   U104^1(tt, V1, V2) -> ACTIVATE(V1)
   U104^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U103^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ACTIVATE(V1)
   U102^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U102^1(tt, V1, V2) -> ACTIVATE(V2)
   U102^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U101^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ACTIVATE(V2)
   ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2)
   U95^1(tt, V2) -> ACTIVATE(V2)
   U94^1(tt, V1, V2) -> ISNAT(activate(V1))
   U94^1(tt, V1, V2) -> ACTIVATE(V1)
   U94^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U93^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ACTIVATE(V1)
   U92^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U92^1(tt, V1, V2) -> ACTIVATE(V2)
   U92^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U91^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ACTIVATE(V2)
   U32^1(tt, V) -> ACTIVATE(V)
   U31^1(tt, V) -> ISNATILISTKIND(activate(V))
   U31^1(tt, V) -> ACTIVATE(V)
   TAKE(s(M), cons(N, IL)) -> U131^1(isNatIList(activate(IL)), activate(IL), M, N)
   U131^1(tt, IL, M, N) -> U132^1(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132^1(tt, IL, M, N) -> U133^1(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133^1(tt, IL, M, N) -> U134^1(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135^1(tt, IL, M, N) -> U136^1(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136^1(tt, IL, M, N) -> ACTIVATE(N)
   U136^1(tt, IL, M, N) -> ACTIVATE(M)
   U136^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ISNATKIND(activate(N))
   U135^1(tt, IL, M, N) -> ACTIVATE(N)
   U135^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ACTIVATE(M)
   U134^1(tt, IL, M, N) -> ISNAT(activate(N))
   U134^1(tt, IL, M, N) -> ACTIVATE(N)
   U134^1(tt, IL, M, N) -> ACTIVATE(IL)
   U134^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ISNATKIND(activate(M))
   U133^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ACTIVATE(IL)
   U133^1(tt, IL, M, N) -> ACTIVATE(N)
   U132^1(tt, IL, M, N) -> ISNAT(activate(M))
   U132^1(tt, IL, M, N) -> ACTIVATE(M)
   U132^1(tt, IL, M, N) -> ACTIVATE(IL)
   U132^1(tt, IL, M, N) -> ACTIVATE(N)
   U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL))
   U131^1(tt, IL, M, N) -> ACTIVATE(IL)
   U131^1(tt, IL, M, N) -> ACTIVATE(M)
   U131^1(tt, IL, M, N) -> ACTIVATE(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)
   U51^1(tt, V2) -> ACTIVATE(V2)
   U121^1(tt, IL) -> ACTIVATE(IL)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2)
   U102^1(tt, V1, V2) -> U103^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104^1(tt, V1, V2) -> U105^1(isNat(activate(V1)), activate(V2))
   U61^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V2)
   U61^1(tt, V2) -> ACTIVATE(V2)
   U104^1(tt, V1, V2) -> ISNAT(activate(V1))
   U104^1(tt, V1, V2) -> ACTIVATE(V1)
   U104^1(tt, V1, V2) -> ACTIVATE(V2)
   U102^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U102^1(tt, V1, V2) -> ACTIVATE(V2)
   U102^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U101^1(tt, V1, V2) -> ACTIVATE(V1)
   U101^1(tt, V1, V2) -> ACTIVATE(V2)
   ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNAT_1(x_1) ) = x_1 + 2
POL( ISNATILIST_1(x_1) ) = x_1 + 2
POL( ISNATILISTKIND_1(x_1) ) = x_1 + 2
POL( ISNATKIND_1(x_1) ) = x_1 + 2
POL( ISNATLIST_1(x_1) ) = x_1 + 2
POL( U101^1_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 1
POL( U102^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 1}
POL( U103^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U104^1_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2
POL( U105^1_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U121^1_2(x_1, x_2) ) = x_1 + x_2
POL( U131^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2
POL( U132^1_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2x_4 + 2
POL( U133^1_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_3 + 2x_4 - 2}
POL( U134^1_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_3 + 2x_4
POL( U135^1_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + x_4
POL( U136^1_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + x_4
POL( U21^1_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U22^1_2(x_1, x_2) ) = x_2 + 2
POL( U31^1_2(x_1, x_2) ) = x_2 + 2
POL( U32^1_2(x_1, x_2) ) = x_2 + 2
POL( U41^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U42^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U43^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U44^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U45^1_2(x_1, x_2) ) = x_2 + 2
POL( U51^1_2(x_1, x_2) ) = x_2 + 2
POL( U61^1_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1}
POL( U91^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U92^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U93^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U94^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U95^1_2(x_1, x_2) ) = x_2 + 2
POL( isNatIList_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( isNatIListKind_1(x_1) ) = x_1
POL( activate_1(x_1) ) = x_1
POL( U106_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = 2x_2 + 1
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1
POL( U46_1(x_1) ) = 2
POL( n__zeros ) = 2
POL( tt ) = 2
POL( n__cons_2(x_1, x_2) ) = x_1 + x_2
POL( U41_3(x_1, ..., x_3) ) = x_1
POL( isNatKind_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -1}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + 2x_4 - 1}
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + x_4 - 1}
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4 + 1
POL( U21_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U22_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U42_3(x_1, ..., x_3) ) = x_1
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = x_1
POL( U51_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U61_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( isNat_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = x_1
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = x_1
POL( U71_1(x_1) ) = 2
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( n__take_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = 2x_1
POL( U81_1(x_1) ) = x_1
POL( zeros ) = 2
POL( take_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__length_1(x_1) ) = 0
POL( n__s_1(x_1) ) = 2x_1
POL( n__nil ) = 2
POL( nil ) = 2
POL( TAKE_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( ACTIVATE_1(x_1) ) = x_1 + 2

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

   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> U121(isNatIList(IL), IL)
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


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

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

   TAKE(0, IL) -> U121^1(isNatIList(IL), IL)
   U121^1(tt, IL) -> ISNATILISTKIND(activate(IL))
   ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2))
   U51^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNATKIND(n__s(V1)) -> ACTIVATE(V1)
   TAKE(0, IL) -> ISNATILIST(IL)
   ISNATILIST(V) -> U31^1(isNatIListKind(activate(V)), activate(V))
   U31^1(tt, V) -> U32^1(isNatIListKind(activate(V)), activate(V))
   U32^1(tt, V) -> ISNATLIST(activate(V))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2))
   U95^1(tt, V2) -> ISNATLIST(activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATLIST(n__take(V1, V2)) -> U101^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101^1(tt, V1, V2) -> U102^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U103^1(tt, V1, V2) -> U104^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U105^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(V) -> ISNATILISTKIND(activate(V))
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2)
   ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2))
   ISNATILIST(V) -> ACTIVATE(V)
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2))
   U45^1(tt, V2) -> ISNATILIST(activate(V2))
   ISNATILIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1))
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1)
   ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2)
   U45^1(tt, V2) -> ACTIVATE(V2)
   U44^1(tt, V1, V2) -> ISNAT(activate(V1))
   ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1))
   U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1))
   U22^1(tt, V1) -> ISNAT(activate(V1))
   ISNAT(n__s(V1)) -> ISNATKIND(activate(V1))
   ISNAT(n__s(V1)) -> ACTIVATE(V1)
   U22^1(tt, V1) -> ACTIVATE(V1)
   U21^1(tt, V1) -> ISNATKIND(activate(V1))
   U21^1(tt, V1) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V1)
   U44^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U43^1(tt, V1, V2) -> ACTIVATE(V2)
   U43^1(tt, V1, V2) -> ACTIVATE(V1)
   U42^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U42^1(tt, V1, V2) -> ACTIVATE(V2)
   U42^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U41^1(tt, V1, V2) -> ACTIVATE(V1)
   U41^1(tt, V1, V2) -> ACTIVATE(V2)
   U105^1(tt, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U103^1(tt, V1, V2) -> ACTIVATE(V2)
   U103^1(tt, V1, V2) -> ACTIVATE(V1)
   U95^1(tt, V2) -> ACTIVATE(V2)
   U94^1(tt, V1, V2) -> ISNAT(activate(V1))
   U94^1(tt, V1, V2) -> ACTIVATE(V1)
   U94^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U93^1(tt, V1, V2) -> ACTIVATE(V2)
   U93^1(tt, V1, V2) -> ACTIVATE(V1)
   U92^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2))
   U92^1(tt, V1, V2) -> ACTIVATE(V2)
   U92^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ISNATKIND(activate(V1))
   U91^1(tt, V1, V2) -> ACTIVATE(V1)
   U91^1(tt, V1, V2) -> ACTIVATE(V2)
   U32^1(tt, V) -> ACTIVATE(V)
   U31^1(tt, V) -> ISNATILISTKIND(activate(V))
   U31^1(tt, V) -> ACTIVATE(V)
   TAKE(s(M), cons(N, IL)) -> U131^1(isNatIList(activate(IL)), activate(IL), M, N)
   U131^1(tt, IL, M, N) -> U132^1(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132^1(tt, IL, M, N) -> U133^1(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133^1(tt, IL, M, N) -> U134^1(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135^1(tt, IL, M, N) -> U136^1(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136^1(tt, IL, M, N) -> ACTIVATE(N)
   U136^1(tt, IL, M, N) -> ACTIVATE(M)
   U136^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ISNATKIND(activate(N))
   U135^1(tt, IL, M, N) -> ACTIVATE(N)
   U135^1(tt, IL, M, N) -> ACTIVATE(IL)
   U135^1(tt, IL, M, N) -> ACTIVATE(M)
   U134^1(tt, IL, M, N) -> ISNAT(activate(N))
   U134^1(tt, IL, M, N) -> ACTIVATE(N)
   U134^1(tt, IL, M, N) -> ACTIVATE(IL)
   U134^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ISNATKIND(activate(M))
   U133^1(tt, IL, M, N) -> ACTIVATE(M)
   U133^1(tt, IL, M, N) -> ACTIVATE(IL)
   U133^1(tt, IL, M, N) -> ACTIVATE(N)
   U132^1(tt, IL, M, N) -> ISNAT(activate(M))
   U132^1(tt, IL, M, N) -> ACTIVATE(M)
   U132^1(tt, IL, M, N) -> ACTIVATE(IL)
   U132^1(tt, IL, M, N) -> ACTIVATE(N)
   U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL))
   U131^1(tt, IL, M, N) -> ACTIVATE(IL)
   U131^1(tt, IL, M, N) -> ACTIVATE(M)
   U131^1(tt, IL, M, N) -> ACTIVATE(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)
   U51^1(tt, V2) -> ACTIVATE(V2)
   U121^1(tt, IL) -> ACTIVATE(IL)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(17) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 94 less nodes.
----------------------------------------

(18)
Complex Obligation (AND)

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

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

   ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ISNATKIND(x_1)) =  	[[5A]] 	 +  	[[4A]] 	* 	x_1
>>>

   <<<
 POL(n__s(x_1)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(activate(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__zeros) =  	[[0A]]
>>>

   <<<
 POL(zeros) =  	[[1A]]
>>>

   <<<
 POL(n__take(x_1, x_2)) =  	[[2A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[3A]] 	* 	x_2
>>>

   <<<
 POL(take(x_1, x_2)) =  	[[2A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[3A]] 	* 	x_2
>>>

   <<<
 POL(n__0) =  	[[2A]]
>>>

   <<<
 POL(0) =  	[[2A]]
>>>

   <<<
 POL(n__length(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(n__cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__nil) =  	[[2A]]
>>>

   <<<
 POL(nil) =  	[[2A]]
>>>

   <<<
 POL(U121(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[3A]] 	* 	x_2
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[2A]]
>>>

   <<<
 POL(U31(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(isNatIListKind(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U111(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

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

   <<<
 POL(isNatKind(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

   <<<
 POL(U61(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U92(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

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

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

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

   <<<
 POL(U11(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U12(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

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

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U105(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

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

   <<<
 POL(U41(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

   <<<
 POL(U43(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U44(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U45(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U46(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

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

   <<<
 POL(U112(x_1, x_2, x_3)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U113(x_1, x_2, x_3)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U114(x_1, x_2)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U32(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U33(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U122(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U131(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[4A]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2 	 +  	[[4A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U132(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2 	 +  	[[4A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U133(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2 	 +  	[[4A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U134(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2 	 +  	[[4A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U135(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2 	 +  	[[4A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U136(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2 	 +  	[[4A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>


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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> U121(isNatIList(IL), IL)
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   U71(tt) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


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

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

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(22) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(23)
YES

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

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1))
   U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1))
   U22^1(tt, V1) -> ISNAT(activate(V1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U22^1(tt, V1) -> ISNAT(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ISNAT(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__s(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(U21^1(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(activate(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[5A]]
>>>

   <<<
 POL(U22^1(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__zeros) =  	[[3A]]
>>>

   <<<
 POL(zeros) =  	[[4A]]
>>>

   <<<
 POL(n__take(x_1, x_2)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(take(x_1, x_2)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__0) =  	[[5A]]
>>>

   <<<
 POL(0) =  	[[5A]]
>>>

   <<<
 POL(n__length(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(n__cons(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__nil) =  	[[5A]]
>>>

   <<<
 POL(nil) =  	[[5A]]
>>>

   <<<
 POL(U71(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(isNatIListKind(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U81(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U121(x_1, x_2)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U111(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(isNatList(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U91(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U92(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U93(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U94(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U95(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U12(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

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

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U105(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

   <<<
 POL(U106(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U41(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U42(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U43(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U44(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U45(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U46(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U96(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U113(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

   <<<
 POL(U32(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U33(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U122(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U131(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U132(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U133(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U134(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U135(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U136(x_1, x_2, x_3, x_4)) =  	[[2A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[3A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>


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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> U121(isNatIList(IL), IL)
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


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

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

   ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1))
   U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(27) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(28)
TRUE

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

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(tt, V2) -> ISNATILISTKIND(activate(V2))
   ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(30) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, V2) -> ISNATILISTKIND(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__zeros) -> ISNATILISTKIND(zeros),U51^1(tt, n__zeros) -> ISNATILISTKIND(zeros))
   (U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1)),U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1)))
   (U51^1(tt, n__0) -> ISNATILISTKIND(0),U51^1(tt, n__0) -> ISNATILISTKIND(0))
   (U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0)),U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0)))
   (U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0)),U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0)))
   (U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1)),U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1)))
   (U51^1(tt, n__nil) -> ISNATILISTKIND(nil),U51^1(tt, n__nil) -> ISNATILISTKIND(nil))
   (U51^1(tt, x0) -> ISNATILISTKIND(x0),U51^1(tt, x0) -> ISNATILISTKIND(x0))


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

(31)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(zeros)
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   U51^1(tt, n__0) -> ISNATILISTKIND(0)
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   U51^1(tt, x0) -> ISNATILISTKIND(x0)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(32) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__zeros, y1)) -> U51^1(isNatKind(zeros), activate(y1)),ISNATILISTKIND(n__cons(n__zeros, y1)) -> U51^1(isNatKind(zeros), activate(y1)))
   (ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1)),ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1)))
   (ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1)),ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1)))
   (ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1)),ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1)))
   (ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1)),ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1)))
   (ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1)),ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1)))
   (ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)),ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)))
   (ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1)),ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1)))


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

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

   U51^1(tt, n__zeros) -> ISNATILISTKIND(zeros)
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   U51^1(tt, n__0) -> ISNATILISTKIND(0)
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(n__zeros, y1)) -> U51^1(isNatKind(zeros), activate(y1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(34) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, n__zeros) -> ISNATILISTKIND(zeros) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)),U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)))
   (U51^1(tt, n__zeros) -> ISNATILISTKIND(n__zeros),U51^1(tt, n__zeros) -> ISNATILISTKIND(n__zeros))


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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   U51^1(tt, n__0) -> ISNATILISTKIND(0)
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(n__zeros, y1)) -> U51^1(isNatKind(zeros), activate(y1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__zeros)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   ISNATILISTKIND(n__cons(n__zeros, y1)) -> U51^1(isNatKind(zeros), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__0) -> ISNATILISTKIND(0)
   ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(38) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__zeros, y1)) -> U51^1(isNatKind(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0)),ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0)))
   (ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__zeros), activate(y0)),ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__zeros), activate(y0)))


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

(39)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__0) -> ISNATILISTKIND(0)
   ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__zeros), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

(41)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1))
   U51^1(tt, n__0) -> ISNATILISTKIND(0)
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(42) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0)),ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0)))


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

(43)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   U51^1(tt, n__0) -> ISNATILISTKIND(0)
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(44) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, n__0) -> ISNATILISTKIND(0) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__0) -> ISNATILISTKIND(n__0),U51^1(tt, n__0) -> ISNATILISTKIND(n__0))


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

(45)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__0) -> ISNATILISTKIND(n__0)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(48) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__s(x0)) -> ISNATILISTKIND(n__s(x0)),U51^1(tt, n__s(x0)) -> ISNATILISTKIND(n__s(x0)))


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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__s(x0)) -> ISNATILISTKIND(n__s(x0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(52) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1)),ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1)))


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

(53)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(54) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1)),U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1)))


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

(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(56) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__cons(x0, x1), y2)) -> U51^1(isNatKind(n__cons(x0, x1)), activate(y2)),ISNATILISTKIND(n__cons(n__cons(x0, x1), y2)) -> U51^1(isNatKind(n__cons(x0, x1)), activate(y2)))


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

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

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__cons(x0, x1), y2)) -> U51^1(isNatKind(n__cons(x0, x1)), activate(y2))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

(59)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(nil)
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(60) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, n__nil) -> ISNATILISTKIND(nil) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__nil) -> ISNATILISTKIND(n__nil),U51^1(tt, n__nil) -> ISNATILISTKIND(n__nil))


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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__nil) -> ISNATILISTKIND(n__nil)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(64) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__nil, y0)) -> U51^1(isNatKind(n__nil), activate(y0)),ISNATILISTKIND(n__cons(n__nil, y0)) -> U51^1(isNatKind(n__nil), activate(y0)))


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

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

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   ISNATILISTKIND(n__cons(n__nil, y0)) -> U51^1(isNatKind(n__nil), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(68) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros)),U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros)))
   (U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros)),U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros)))


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

(69)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(70) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__cons(0, n__zeros)), activate(y0)),ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__cons(0, n__zeros)), activate(y0)))
   (ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(n__0, n__zeros)), activate(y0)),ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(n__0, n__zeros)), activate(y0)))


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

(71)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__cons(0, n__zeros)), activate(y0))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

(73)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(n__0, n__zeros)), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(74) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__cons(n__0, n__zeros)), activate(y0)),ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__cons(n__0, n__zeros)), activate(y0)))


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

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

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros))
   ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(78) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros)),U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros)))


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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(x0, x1)), activate(y1))
   ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATILISTKIND_1(x_1) ) = 2x_1 + 2
POL( U51^1_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( take_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( 0 ) = 0
POL( U121_2(x_1, x_2) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 1
POL( s_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + x_3 + 2x_4 - 1}
POL( activate_1(x_1) ) = x_1
POL( n__take_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( U101_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U102_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U114_2(x_1, x_2) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 + 1
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 + 1
POL( U21_2(x_1, x_2) ) = 2x_2 + 2
POL( U22_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U51_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( U81_1(x_1) ) = 1
POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U92_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( isNatKind_1(x_1) ) = 1
POL( n__0 ) = 0
POL( tt ) = 1
POL( n__length_1(x_1) ) = 2
POL( U71_1(x_1) ) = 1
POL( isNatIListKind_1(x_1) ) = 1
POL( n__s_1(x_1) ) = x_1
POL( U103_3(x_1, ..., x_3) ) = x_1 + 2
POL( U104_3(x_1, ..., x_3) ) = x_1 + 1
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U132_4(x_1, ..., x_4) ) = x_1 + 2x_2 + x_3 + 2x_4
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 + 1
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 + 1
POL( U31_2(x_1, x_2) ) = 1
POL( U32_2(x_1, x_2) ) = 1
POL( U43_3(x_1, ..., x_3) ) = x_1
POL( U44_3(x_1, ..., x_3) ) = x_1
POL( U45_2(x_1, x_2) ) = 1
POL( U93_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 2x_2
POL( U95_2(x_1, x_2) ) = 1
POL( isNat_1(x_1) ) = 2x_1 + 2
POL( U106_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 2x_1 + 2
POL( U122_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = x_1
POL( U33_1(x_1) ) = 1
POL( U46_1(x_1) ) = max{0, 2x_1 - 1}
POL( U52_1(x_1) ) = max{0, 2x_1 - 1}
POL( U62_1(x_1) ) = 1
POL( U96_1(x_1) ) = 1
POL( length_1(x_1) ) = 2
POL( n__zeros ) = 1
POL( zeros ) = 1
POL( n__cons_2(x_1, x_2) ) = x_1 + x_2
POL( n__nil ) = 0
POL( nil ) = 0

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

   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

(81)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATILISTKIND_1(x_1) ) = 2x_1 + 1
POL( U51^1_2(x_1, x_2) ) = 2x_2 + 1
POL( take_2(x_1, x_2) ) = 2x_2 + 1
POL( 0 ) = 0
POL( U121_2(x_1, x_2) ) = 1
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( s_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1
POL( activate_1(x_1) ) = x_1
POL( n__take_2(x_1, x_2) ) = 2x_2 + 1
POL( length_1(x_1) ) = 2x_1 + 1
POL( nil ) = 1
POL( U111_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1
POL( isNatList_1(x_1) ) = 2x_1 + 1
POL( n__length_1(x_1) ) = 2x_1 + 1
POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U102_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U114_2(x_1, x_2) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U51_2(x_1, x_2) ) = x_2
POL( U61_2(x_1, x_2) ) = x_2
POL( U81_1(x_1) ) = 1
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = 1
POL( isNatKind_1(x_1) ) = max{0, x_1 - 1}
POL( n__0 ) = 0
POL( tt ) = 1
POL( U71_1(x_1) ) = x_1
POL( isNatIListKind_1(x_1) ) = x_1
POL( n__s_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U104_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U105_2(x_1, x_2) ) = 2x_2 + 2
POL( U11_2(x_1, x_2) ) = 2x_2 + 1
POL( U112_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 1
POL( U113_3(x_1, ..., x_3) ) = x_1 + 2
POL( U12_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1
POL( U31_2(x_1, x_2) ) = x_1 + 1
POL( U32_2(x_1, x_2) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U94_3(x_1, ..., x_3) ) = 1
POL( U95_2(x_1, x_2) ) = 1
POL( isNat_1(x_1) ) = x_1 + 1
POL( U106_1(x_1) ) = x_1 + 1
POL( U122_1(x_1) ) = 1
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 1
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = x_1
POL( U96_1(x_1) ) = 1
POL( n__zeros ) = 1
POL( zeros ) = 1
POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( n__nil ) = 1

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

   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

(83)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(x0))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATILISTKIND_1(x_1) ) = x_1 + 1
POL( U51^1_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 1}
POL( take_2(x_1, x_2) ) = 0
POL( 0 ) = 2
POL( U121_2(x_1, x_2) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( s_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2x_2
POL( U131_4(x_1, ..., x_4) ) = 0
POL( activate_1(x_1) ) = x_1
POL( n__take_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = x_1 + 2
POL( nil ) = 0
POL( U111_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( isNatList_1(x_1) ) = 2x_1 + 1
POL( n__length_1(x_1) ) = x_1 + 2
POL( U101_3(x_1, ..., x_3) ) = x_1
POL( U102_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_1 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 - 2}
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = x_1
POL( U51_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( U81_1(x_1) ) = 1
POL( U91_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( isNatKind_1(x_1) ) = 1
POL( n__0 ) = 2
POL( tt ) = 1
POL( U71_1(x_1) ) = 1
POL( isNatIListKind_1(x_1) ) = x_1 + 2
POL( n__s_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 1
POL( U104_3(x_1, ..., x_3) ) = 1
POL( U105_2(x_1, x_2) ) = x_1
POL( U11_2(x_1, x_2) ) = 1
POL( U112_3(x_1, ..., x_3) ) = 2x_2
POL( U113_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U12_2(x_1, x_2) ) = 1
POL( U132_4(x_1, ..., x_4) ) = max{0, -2}
POL( U133_4(x_1, ..., x_4) ) = 0
POL( U135_4(x_1, ..., x_4) ) = 0
POL( U31_2(x_1, x_2) ) = x_1
POL( U32_2(x_1, x_2) ) = 1
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U95_2(x_1, x_2) ) = 2x_2 + 1
POL( isNat_1(x_1) ) = 1
POL( U106_1(x_1) ) = 1
POL( U122_1(x_1) ) = 0
POL( U13_1(x_1) ) = 1
POL( U23_1(x_1) ) = 1
POL( U33_1(x_1) ) = 1
POL( U46_1(x_1) ) = 1
POL( U52_1(x_1) ) = 1
POL( U62_1(x_1) ) = 1
POL( U96_1(x_1) ) = x_1
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( n__cons_2(x_1, x_2) ) = 2x_2
POL( n__nil ) = 0

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

   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   0 -> n__0
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

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

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(x0, x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(U51^1(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2
>>>

   <<<
 POL(tt) =  	[[2A]]
>>>

   <<<
 POL(n__take(x_1, x_2)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(ISNATILISTKIND(x_1)) =  	[[5A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(take(x_1, x_2)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(n__cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(activate(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__0) =  	[[4A]]
>>>

   <<<
 POL(n__zeros) =  	[[0A]]
>>>

   <<<
 POL(0) =  	[[4A]]
>>>

   <<<
 POL(U121(x_1, x_2)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U131(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(n__length(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(isNatIListKind(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(n__s(x_1)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(U81(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(zeros) =  	[[1A]]
>>>

   <<<
 POL(length(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__nil) =  	[[4A]]
>>>

   <<<
 POL(nil) =  	[[4A]]
>>>

   <<<
 POL(U31(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U111(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

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

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

   <<<
 POL(U51(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

   <<<
 POL(U92(x_1, x_2, x_3)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

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

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

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

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

   <<<
 POL(U11(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U12(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U101(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U105(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(U106(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

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

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

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

   <<<
 POL(U44(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U45(x_1, x_2)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U46(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

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

   <<<
 POL(U112(x_1, x_2, x_3)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U113(x_1, x_2, x_3)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U114(x_1, x_2)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U32(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U33(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U122(x_1)) =  	[[2A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(U132(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U133(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U134(x_1, x_2, x_3, x_4)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U135(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U136(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[2A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>


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

   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

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

   U51^1(tt, x0) -> ISNATILISTKIND(x0)
   ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))
   U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1))
   ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(0, n__zeros))
   U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(88) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U51^1(isNatKind(n__0), activate(n__zeros)) evaluates to  t =U51^1(isNatKind(n__0), activate(n__zeros))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

U51^1(isNatKind(n__0), activate(n__zeros)) -> U51^1(isNatKind(n__0), n__zeros)
with rule activate(X) -> X at position [1] and matcher [X / n__zeros]

U51^1(isNatKind(n__0), n__zeros) -> U51^1(tt, n__zeros)
with rule isNatKind(n__0) -> tt at position [0] and matcher [ ]

U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros))
with rule U51^1(tt, n__zeros) -> ISNATILISTKIND(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATILISTKIND(n__cons(n__0, n__zeros)) -> U51^1(isNatKind(n__0), activate(n__zeros))
with rule ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




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

(89)
NO

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

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

   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2))
   U95^1(tt, V2) -> ISNATLIST(activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(91) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__zeros, y1) -> U95^1(isNat(zeros), activate(y1)),U94^1(tt, n__zeros, y1) -> U95^1(isNat(zeros), activate(y1)))
   (U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1)),U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1)))
   (U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1)),U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1)))
   (U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1)),U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1)))
   (U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1)),U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1)))
   (U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1)),U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1)))
   (U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1)),U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1)))
   (U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1)),U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1)))


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

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

   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, V2) -> ISNATLIST(activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__zeros, y1) -> U95^1(isNat(zeros), activate(y1))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(93) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, V2) -> ISNATLIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__zeros) -> ISNATLIST(zeros),U95^1(tt, n__zeros) -> ISNATLIST(zeros))
   (U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1)),U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1)))
   (U95^1(tt, n__0) -> ISNATLIST(0),U95^1(tt, n__0) -> ISNATLIST(0))
   (U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0)),U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0)))
   (U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0)),U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0)))
   (U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)),U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)))
   (U95^1(tt, n__nil) -> ISNATLIST(nil),U95^1(tt, n__nil) -> ISNATLIST(nil))
   (U95^1(tt, x0) -> ISNATLIST(x0),U95^1(tt, x0) -> ISNATLIST(x0))


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

(94)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__zeros, y1) -> U95^1(isNat(zeros), activate(y1))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U95^1(tt, n__zeros) -> ISNATLIST(zeros)
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__0) -> ISNATLIST(0)
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(95) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, n__zeros, y1) -> U95^1(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0)),U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0)))
   (U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__zeros), activate(y0)),U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__zeros), activate(y0)))


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

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

   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U95^1(tt, n__zeros) -> ISNATLIST(zeros)
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__0) -> ISNATLIST(0)
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__zeros), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__zeros) -> ISNATLIST(zeros)
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__0) -> ISNATLIST(0)
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(99) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, n__zeros) -> ISNATLIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)),U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)))
   (U95^1(tt, n__zeros) -> ISNATLIST(n__zeros),U95^1(tt, n__zeros) -> ISNATLIST(n__zeros))


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

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

   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__0) -> ISNATLIST(0)
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__zeros)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__0) -> ISNATLIST(0)
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(103) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, n__0) -> ISNATLIST(0) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__0) -> ISNATLIST(n__0),U95^1(tt, n__0) -> ISNATLIST(n__0))


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

(104)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__0) -> ISNATLIST(n__0)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

(106)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(107) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__s(x0)) -> ISNATLIST(n__s(x0)),U95^1(tt, n__s(x0)) -> ISNATLIST(n__s(x0)))


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

(108)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__s(x0)) -> ISNATLIST(n__s(x0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(111) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)),U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)))


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

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

   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, n__nil) -> ISNATLIST(nil)
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(113) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, n__nil) -> ISNATLIST(nil) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__nil) -> ISNATLIST(n__nil),U95^1(tt, n__nil) -> ISNATLIST(n__nil))


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

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

   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__nil) -> ISNATLIST(n__nil)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

(116)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(117) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)),U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)))
   (U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)),U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)))


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

(118)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(119) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0)),U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0)))


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

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

   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(121) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1)),U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1)))


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

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

   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(123) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__cons(x0, x1), y2) -> U95^1(isNat(n__cons(x0, x1)), activate(y2)),U94^1(tt, n__cons(x0, x1), y2) -> U95^1(isNat(n__cons(x0, x1)), activate(y2)))


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

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

   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))
   U94^1(tt, n__cons(x0, x1), y2) -> U95^1(isNat(n__cons(x0, x1)), activate(y2))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(127) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U95^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)),U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)))


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

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

   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(129) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, n__nil, y1) -> U95^1(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__nil, y0) -> U95^1(isNat(n__nil), activate(y0)),U94^1(tt, n__nil, y0) -> U95^1(isNat(n__nil), activate(y0)))


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

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

   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, n__nil, y0) -> U95^1(isNat(n__nil), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(133) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__cons(0, n__zeros)), activate(y0)),U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__cons(0, n__zeros)), activate(y0)))
   (U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(n__0, n__zeros)), activate(y0)),U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(n__0, n__zeros)), activate(y0)))


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

(134)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__cons(0, n__zeros)), activate(y0))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

(136)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(137) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U94^1(tt, n__zeros, y0) -> U95^1(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

   (U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__cons(n__0, n__zeros)), activate(y0)),U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__cons(n__0, n__zeros)), activate(y0)))


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

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

   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))
   U94^1(tt, n__zeros, y0) -> U95^1(isNat(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(x0, x1)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U91^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U92^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U93^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U94^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U95^1_2(x_1, x_2) ) = 2x_2 + 2
POL( ISNATLIST_1(x_1) ) = 2x_1 + 2
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1}
POL( U102_3(x_1, ..., x_3) ) = x_1 + 1
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U112_3(x_1, ..., x_3) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 0
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = x_1 + 2x_2 + x_4
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = x_1
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U42_3(x_1, ..., x_3) ) = x_1
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 1}
POL( U93_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U95_2(x_1, x_2) ) = 2x_2 + 1
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 2x_1 + 1
POL( U122_1(x_1) ) = 2
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2
POL( U96_1(x_1) ) = x_1
POL( n__take_2(x_1, x_2) ) = 2x_2 + 2
POL( length_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = 2x_1
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 2
POL( zeros ) = 2
POL( take_2(x_1, x_2) ) = 2x_2 + 2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__length_1(x_1) ) = 0
POL( n__s_1(x_1) ) = 2x_1
POL( n__cons_2(x_1, x_2) ) = x_1 + x_2
POL( n__nil ) = 2
POL( nil ) = 2
POL( tt ) = 2
POL( U121_2(x_1, x_2) ) = x_1

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

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

   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(x0)), activate(y1))
   U94^1(tt, n__s(x0), y1) -> U95^1(isNat(n__s(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U91^1_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3
POL( U92^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U93^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U94^1_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3
POL( U95^1_2(x_1, x_2) ) = 2x_2 + 2
POL( ISNATLIST_1(x_1) ) = 2x_1 + 2
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, x_1 - 1}
POL( U112_3(x_1, ..., x_3) ) = max{0, x_1 - 1}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = 1
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_4 - 2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = x_1
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( n__take_2(x_1, x_2) ) = x_2 + 2
POL( length_1(x_1) ) = 1
POL( s_1(x_1) ) = 1
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( take_2(x_1, x_2) ) = x_2 + 2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__length_1(x_1) ) = 1
POL( n__s_1(x_1) ) = 1
POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( n__nil ) = 0
POL( nil ) = 0
POL( tt ) = 2
POL( U121_2(x_1, x_2) ) = 2

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   0 -> n__0
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

(144)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U95^1(tt, n__length(x0)) -> ISNATLIST(length(x0))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U91^1_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U92^1_3(x_1, ..., x_3) ) = x_1 + 2x_3
POL( U93^1_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U94^1_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U95^1_2(x_1, x_2) ) = 2x_2 + 1
POL( ISNATLIST_1(x_1) ) = x_1 + 1
POL( U101_3(x_1, ..., x_3) ) = 2x_1
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 1
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 0
POL( U132_4(x_1, ..., x_4) ) = 0
POL( U133_4(x_1, ..., x_4) ) = max{0, -2}
POL( U134_4(x_1, ..., x_4) ) = 0
POL( U135_4(x_1, ..., x_4) ) = max{0, -2}
POL( U136_4(x_1, ..., x_4) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( U22_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 1
POL( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = x_1
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( U91_3(x_1, ..., x_3) ) = x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = x_1
POL( U95_2(x_1, x_2) ) = 1
POL( cons_2(x_1, x_2) ) = 2x_2
POL( isNatKind_1(x_1) ) = 1
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = x_1
POL( U106_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = 0
POL( U13_1(x_1) ) = x_1
POL( U23_1(x_1) ) = x_1
POL( U33_1(x_1) ) = 1
POL( U46_1(x_1) ) = 1
POL( U52_1(x_1) ) = 1
POL( U62_1(x_1) ) = 1
POL( U71_1(x_1) ) = 1
POL( U81_1(x_1) ) = 1
POL( U96_1(x_1) ) = 1
POL( n__take_2(x_1, x_2) ) = max{0, -1}
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( take_2(x_1, x_2) ) = 0
POL( n__0 ) = 1
POL( 0 ) = 1
POL( n__length_1(x_1) ) = 2
POL( n__s_1(x_1) ) = x_1
POL( n__cons_2(x_1, x_2) ) = 2x_2
POL( n__nil ) = 0
POL( nil ) = 0
POL( tt ) = 1
POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2}

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   0 -> n__0
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

(146)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(x0, x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ISNATLIST(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__cons(x_1, x_2)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U91^1(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(activate(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[4A]]
>>>

   <<<
 POL(U92^1(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U93^1(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(isNatIListKind(x_1)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(U94^1(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U95^1(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__take(x_1, x_2)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(take(x_1, x_2)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(n__zeros) =  	[[2A]]
>>>

   <<<
 POL(0) =  	[[4A]]
>>>

   <<<
 POL(n__0) =  	[[4A]]
>>>

   <<<
 POL(isNat(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(zeros) =  	[[3A]]
>>>

   <<<
 POL(n__length(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__s(x_1)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__nil) =  	[[4A]]
>>>

   <<<
 POL(nil) =  	[[4A]]
>>>

   <<<
 POL(U71(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U81(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U121(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U131(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U11(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

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

   <<<
 POL(U31(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U111(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(isNatList(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U91(x_1, x_2, x_3)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U52(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U62(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U92(x_1, x_2, x_3)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U93(x_1, x_2, x_3)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U94(x_1, x_2, x_3)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U95(x_1, x_2)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U12(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U101(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U105(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U106(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U41(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U42(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U43(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U44(x_1, x_2, x_3)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U45(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U46(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U96(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U112(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U113(x_1, x_2, x_3)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3
>>>

   <<<
 POL(U114(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U32(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U33(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U122(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U132(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U133(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U134(x_1, x_2, x_3, x_4)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U135(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>

   <<<
 POL(U136(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3 	 +  	[[-I]] 	* 	x_4
>>>


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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

(148)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U91^1(tt, V1, V2) -> U92^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92^1(tt, V1, V2) -> U93^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93^1(tt, V1, V2) -> U94^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U95^1(tt, x0) -> ISNATLIST(x0)
   U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))
   U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))
   U94^1(tt, x0, y1) -> U95^1(isNat(x0), activate(y1))
   U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2))
   U45^1(tt, V2) -> ISNATILIST(activate(V2))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(150) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__zeros, y1) -> U45^1(isNat(zeros), activate(y1)),U44^1(tt, n__zeros, y1) -> U45^1(isNat(zeros), activate(y1)))
   (U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1)),U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1)))
   (U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1)),U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1)))
   (U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1)),U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1)))
   (U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1)),U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1)))
   (U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1)),U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1)))
   (U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1)),U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1)))
   (U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)),U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)))


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

(151)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, V2) -> ISNATILIST(activate(V2))
   U44^1(tt, n__zeros, y1) -> U45^1(isNat(zeros), activate(y1))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(152) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, V2) -> ISNATILIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__zeros) -> ISNATILIST(zeros),U45^1(tt, n__zeros) -> ISNATILIST(zeros))
   (U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1)),U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1)))
   (U45^1(tt, n__0) -> ISNATILIST(0),U45^1(tt, n__0) -> ISNATILIST(0))
   (U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0)),U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0)))
   (U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0)),U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0)))
   (U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)),U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)))
   (U45^1(tt, n__nil) -> ISNATILIST(nil),U45^1(tt, n__nil) -> ISNATILIST(nil))
   (U45^1(tt, x0) -> ISNATILIST(x0),U45^1(tt, x0) -> ISNATILIST(x0))


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

(153)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__zeros, y1) -> U45^1(isNat(zeros), activate(y1))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U45^1(tt, n__zeros) -> ISNATILIST(zeros)
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   U45^1(tt, n__0) -> ISNATILIST(0)
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(154) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, n__zeros, y1) -> U45^1(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0)),U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0)))
   (U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__zeros), activate(y0)),U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__zeros), activate(y0)))


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

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

   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U45^1(tt, n__zeros) -> ISNATILIST(zeros)
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   U45^1(tt, n__0) -> ISNATILIST(0)
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__zeros), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__zeros) -> ISNATILIST(zeros)
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   U45^1(tt, n__0) -> ISNATILIST(0)
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(158) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, n__zeros) -> ISNATILIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)),U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)))
   (U45^1(tt, n__zeros) -> ISNATILIST(n__zeros),U45^1(tt, n__zeros) -> ISNATILIST(n__zeros))


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

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

   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   U45^1(tt, n__0) -> ISNATILIST(0)
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__zeros)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__0) -> ISNATILIST(0)
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(162) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, n__0) -> ISNATILIST(0) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__0) -> ISNATILIST(n__0),U45^1(tt, n__0) -> ISNATILIST(n__0))


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

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

   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U45^1(tt, n__0) -> ISNATILIST(n__0)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(166) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__s(x0)) -> ISNATILIST(n__s(x0)),U45^1(tt, n__s(x0)) -> ISNATILIST(n__s(x0)))


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

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

   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U45^1(tt, n__s(x0)) -> ISNATILIST(n__s(x0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(170) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, n__0, y1) -> U45^1(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)),U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)))


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

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

   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(172) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)),U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)))


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

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

   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, n__nil) -> ISNATILIST(nil)
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(174) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, n__nil) -> ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__nil) -> ISNATILIST(n__nil),U45^1(tt, n__nil) -> ISNATILIST(n__nil))


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

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

   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__nil) -> ISNATILIST(n__nil)

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

Q is empty.
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:

   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(178) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)),U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)))
   (U45^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)),U45^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)))


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

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

   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(180) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1)),U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1)))


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

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

   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(182) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__cons(x0, x1), y2) -> U45^1(isNat(n__cons(x0, x1)), activate(y2)),U44^1(tt, n__cons(x0, x1), y2) -> U45^1(isNat(n__cons(x0, x1)), activate(y2)))


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

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

   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))
   U44^1(tt, n__cons(x0, x1), y2) -> U45^1(isNat(n__cons(x0, x1)), activate(y2))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

Q is empty.
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:

   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(186) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U45^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)),U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)))


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

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

   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(188) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, n__nil, y1) -> U45^1(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__nil, y0) -> U45^1(isNat(n__nil), activate(y0)),U44^1(tt, n__nil, y0) -> U45^1(isNat(n__nil), activate(y0)))


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

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

   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, n__nil, y0) -> U45^1(isNat(n__nil), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

Q is empty.
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:

   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(192) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__cons(0, n__zeros)), activate(y0)),U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__cons(0, n__zeros)), activate(y0)))
   (U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(n__0, n__zeros)), activate(y0)),U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(n__0, n__zeros)), activate(y0)))


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

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

   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__cons(0, n__zeros)), activate(y0))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

(196) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U44^1(tt, n__zeros, y0) -> U45^1(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

   (U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__cons(n__0, n__zeros)), activate(y0)),U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__cons(n__0, n__zeros)), activate(y0)))


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

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

   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))
   U44^1(tt, n__zeros, y0) -> U45^1(isNat(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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

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

   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))
   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U44^1(tt, n__s(x0), y1) -> U45^1(isNat(n__s(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U41^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3
POL( U42^1_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U43^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3
POL( U44^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3
POL( U45^1_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( ISNATILIST_1(x_1) ) = 2x_1
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2}
POL( U102_3(x_1, ..., x_3) ) = 2x_3
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = 2x_3
POL( U105_2(x_1, x_2) ) = 2x_2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, x_1 - 1}
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = 1
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_4
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_4
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_4
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_4
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_4
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_4
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = x_2
POL( U32_2(x_1, x_2) ) = x_2
POL( U41_3(x_1, ..., x_3) ) = x_2 + x_3 + 1
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 1}
POL( U43_3(x_1, ..., x_3) ) = x_2 + x_3 + 1
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 1}
POL( U45_2(x_1, x_2) ) = x_2 + 1
POL( U51_2(x_1, x_2) ) = x_1
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = x_3
POL( U93_3(x_1, ..., x_3) ) = x_3
POL( U94_3(x_1, ..., x_3) ) = x_3
POL( U95_2(x_1, x_2) ) = x_2
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = x_1 + 2
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = x_1
POL( U122_1(x_1) ) = 2
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = x_1
POL( U46_1(x_1) ) = x_1
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2
POL( U96_1(x_1) ) = x_1
POL( n__take_2(x_1, x_2) ) = 2x_2
POL( length_1(x_1) ) = x_1
POL( s_1(x_1) ) = 1
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 2
POL( zeros ) = 2
POL( take_2(x_1, x_2) ) = 2x_2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__length_1(x_1) ) = x_1
POL( n__s_1(x_1) ) = 1
POL( n__cons_2(x_1, x_2) ) = x_1 + x_2
POL( n__nil ) = 2
POL( nil ) = 2
POL( tt ) = 2
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 - 2}

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

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

   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U41^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1
POL( U42^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1
POL( U43^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1
POL( U44^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1
POL( U45^1_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( ISNATILIST_1(x_1) ) = 2x_1 + 1
POL( U101_3(x_1, ..., x_3) ) = 2x_3
POL( U102_3(x_1, ..., x_3) ) = 2x_3
POL( U103_3(x_1, ..., x_3) ) = 2x_3
POL( U104_3(x_1, ..., x_3) ) = x_1
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2x_2 + 2
POL( U111_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 1
POL( U112_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 1
POL( U113_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 1}
POL( U114_2(x_1, x_2) ) = 2x_2 + 1
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2x_2
POL( U61_2(x_1, x_2) ) = 2x_2
POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = x_1 + 2x_3
POL( U93_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U94_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U95_2(x_1, x_2) ) = 2x_2 + 2
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( isNatKind_1(x_1) ) = 2x_1 + 2
POL( isNatIListKind_1(x_1) ) = 2x_1
POL( isNat_1(x_1) ) = x_1 + 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2x_1 + 2
POL( U122_1(x_1) ) = x_1
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = x_1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2
POL( U96_1(x_1) ) = 2
POL( n__take_2(x_1, x_2) ) = x_1 + 2x_2
POL( length_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = x_1
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 1
POL( zeros ) = 1
POL( take_2(x_1, x_2) ) = x_1 + 2x_2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__length_1(x_1) ) = 2x_1 + 1
POL( n__s_1(x_1) ) = x_1
POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( n__nil ) = 2
POL( nil ) = 2
POL( tt ) = 2
POL( U121_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

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

   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U45^1(tt, n__length(x0)) -> ISNATILIST(length(x0))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U41^1_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U42^1_3(x_1, ..., x_3) ) = x_1 + 2x_3
POL( U43^1_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U44^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U45^1_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( ISNATILIST_1(x_1) ) = x_1 + 2
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, -2}
POL( U134_4(x_1, ..., x_4) ) = max{0, -2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_1 - 2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = x_1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = x_1
POL( U94_3(x_1, ..., x_3) ) = x_1
POL( U95_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = x_1 + 2
POL( U122_1(x_1) ) = max{0, -1}
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2
POL( U96_1(x_1) ) = 2
POL( n__take_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 0
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( take_2(x_1, x_2) ) = 0
POL( n__0 ) = 1
POL( 0 ) = 1
POL( n__length_1(x_1) ) = 2
POL( n__s_1(x_1) ) = 0
POL( n__cons_2(x_1, x_2) ) = 2x_2
POL( n__nil ) = 0
POL( nil ) = 0
POL( tt ) = 2
POL( U121_2(x_1, x_2) ) = 0

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   length(X) -> n__length(X)
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

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

   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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

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


The following pairs can be oriented strictly and are deleted.

   U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(x0, x1)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U41^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3
POL( U42^1_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U43^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3
POL( U44^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3
POL( U45^1_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( ISNATILIST_1(x_1) ) = 2x_1
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_4
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = x_1 + 2x_2 + x_4
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( U61_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = x_1 + 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = x_1
POL( U13_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = x_1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2
POL( U96_1(x_1) ) = 2
POL( n__take_2(x_1, x_2) ) = 2x_2 + 2
POL( length_1(x_1) ) = x_1 + 2
POL( s_1(x_1) ) = 2
POL( activate_1(x_1) ) = x_1
POL( n__zeros ) = 2
POL( zeros ) = 2
POL( take_2(x_1, x_2) ) = 2x_2 + 2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__length_1(x_1) ) = x_1 + 2
POL( n__s_1(x_1) ) = 2
POL( n__cons_2(x_1, x_2) ) = x_1 + x_2
POL( n__nil ) = 2
POL( nil ) = 2
POL( tt ) = 2
POL( U121_2(x_1, x_2) ) = 2x_2 + 1

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

   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   take(X1, X2) -> n__take(X1, X2)
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   0 -> n__0
   isNatIList(n__zeros) -> tt
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   length(nil) -> 0
   length(X) -> n__length(X)
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   U71(tt) -> tt
   U81(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U13(tt) -> tt
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   s(X) -> n__s(X)
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   cons(X1, X2) -> n__cons(X1, X2)
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   nil -> n__nil


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

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

   U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1))
   ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U45^1(tt, x0) -> ISNATILIST(x0)
   U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))
   U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))
   U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1))
   U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2))
   U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2))
   U105(tt, V2) -> U106(isNatIList(activate(V2)))
   U106(tt) -> tt
   U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1))
   U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N))
   U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N))
   U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L))
   U114(tt, L) -> s(length(activate(L)))
   U12(tt, V1) -> U13(isNatList(activate(V1)))
   U121(tt, IL) -> U122(isNatIListKind(activate(IL)))
   U122(tt) -> nil
   U13(tt) -> tt
   U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
   U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
   U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
   U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
   U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
   U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL)))
   U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1))
   U22(tt, V1) -> U23(isNat(activate(V1)))
   U23(tt) -> tt
   U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V))
   U32(tt, V) -> U33(isNatList(activate(V)))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2))
   U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2))
   U45(tt, V2) -> U46(isNatIList(activate(V2)))
   U46(tt) -> tt
   U51(tt, V2) -> U52(isNatIListKind(activate(V2)))
   U52(tt) -> tt
   U61(tt, V2) -> U62(isNatIListKind(activate(V2)))
   U62(tt) -> tt
   U71(tt) -> tt
   U81(tt) -> tt
   U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2))
   U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
   U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2))
   U95(tt, V2) -> U96(isNatList(activate(V2)))
   U96(tt) -> tt
   isNat(n__0) -> tt
   isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
   isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatIListKind(n__nil) -> tt
   isNatIListKind(n__zeros) -> tt
   isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2))
   isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2))
   isNatKind(n__0) -> tt
   isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1)))
   isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2))
   isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2))
   length(nil) -> 0
   length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N)
   take(0, IL) -> U121(isNatIList(IL), IL)
   take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N)
   zeros -> n__zeros
   take(X1, X2) -> n__take(X1, X2)
   0 -> n__0
   length(X) -> n__length(X)
   s(X) -> n__s(X)
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   activate(n__zeros) -> zeros
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__0) -> 0
   activate(n__length(X)) -> length(X)
   activate(n__s(X)) -> s(X)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__nil) -> nil
   activate(X) -> X

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