/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- NO proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 68 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 4909 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) QDP (10) QDPOrderProof [EQUIVALENT, 1434 ms] (11) QDP (12) DependencyGraphProof [EQUIVALENT, 0 ms] (13) AND (14) QDP (15) QDPOrderProof [EQUIVALENT, 1046 ms] (16) QDP (17) PisEmptyProof [EQUIVALENT, 0 ms] (18) YES (19) QDP (20) QDPOrderProof [EQUIVALENT, 1451 ms] (21) QDP (22) DependencyGraphProof [EQUIVALENT, 0 ms] (23) TRUE (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) DependencyGraphProof [EQUIVALENT, 0 ms] (32) QDP (33) TransformationProof [EQUIVALENT, 0 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) QDP (37) TransformationProof [EQUIVALENT, 0 ms] (38) QDP (39) DependencyGraphProof [EQUIVALENT, 0 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 0 ms] (44) QDP (45) DependencyGraphProof [EQUIVALENT, 0 ms] (46) QDP (47) TransformationProof [EQUIVALENT, 0 ms] (48) QDP (49) TransformationProof [EQUIVALENT, 0 ms] (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) DependencyGraphProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) DependencyGraphProof [EQUIVALENT, 0 ms] (62) QDP (63) TransformationProof [EQUIVALENT, 1 ms] (64) QDP (65) DependencyGraphProof [EQUIVALENT, 0 ms] (66) QDP (67) TransformationProof [EQUIVALENT, 0 ms] (68) QDP (69) TransformationProof [EQUIVALENT, 0 ms] (70) QDP (71) DependencyGraphProof [EQUIVALENT, 0 ms] (72) QDP (73) TransformationProof [EQUIVALENT, 0 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 706 ms] (76) QDP (77) QDPOrderProof [EQUIVALENT, 492 ms] (78) QDP (79) QDPOrderProof [EQUIVALENT, 545 ms] (80) QDP (81) QDPOrderProof [EQUIVALENT, 2298 ms] (82) QDP (83) NonTerminationLoopProof [COMPLETE, 105 ms] (84) NO (85) QDP (86) TransformationProof [EQUIVALENT, 0 ms] (87) QDP (88) TransformationProof [EQUIVALENT, 0 ms] (89) QDP (90) TransformationProof [EQUIVALENT, 0 ms] (91) QDP (92) DependencyGraphProof [EQUIVALENT, 0 ms] (93) QDP (94) TransformationProof [EQUIVALENT, 0 ms] (95) QDP (96) DependencyGraphProof [EQUIVALENT, 0 ms] (97) QDP (98) TransformationProof [EQUIVALENT, 0 ms] (99) QDP (100) DependencyGraphProof [EQUIVALENT, 0 ms] (101) QDP (102) TransformationProof [EQUIVALENT, 0 ms] (103) QDP (104) DependencyGraphProof [EQUIVALENT, 0 ms] (105) QDP (106) TransformationProof [EQUIVALENT, 0 ms] (107) QDP (108) TransformationProof [EQUIVALENT, 0 ms] (109) QDP (110) DependencyGraphProof [EQUIVALENT, 0 ms] (111) QDP (112) TransformationProof [EQUIVALENT, 0 ms] (113) QDP (114) TransformationProof [EQUIVALENT, 0 ms] (115) QDP (116) TransformationProof [EQUIVALENT, 0 ms] (117) QDP (118) TransformationProof [EQUIVALENT, 0 ms] (119) QDP (120) DependencyGraphProof [EQUIVALENT, 0 ms] (121) QDP (122) TransformationProof [EQUIVALENT, 0 ms] (123) QDP (124) TransformationProof [EQUIVALENT, 0 ms] (125) QDP (126) DependencyGraphProof [EQUIVALENT, 0 ms] (127) QDP (128) TransformationProof [EQUIVALENT, 0 ms] (129) QDP (130) DependencyGraphProof [EQUIVALENT, 0 ms] (131) QDP (132) TransformationProof [EQUIVALENT, 0 ms] (133) QDP (134) DependencyGraphProof [EQUIVALENT, 0 ms] (135) QDP (136) QDPOrderProof [EQUIVALENT, 2417 ms] (137) QDP (138) QDPOrderProof [EQUIVALENT, 1831 ms] (139) QDP (140) QDPOrderProof [EQUIVALENT, 1849 ms] (141) QDP (142) QDPOrderProof [EQUIVALENT, 2081 ms] (143) QDP (144) QDP (145) QDPOrderProof [EQUIVALENT, 1515 ms] (146) QDP (147) DependencyGraphProof [EQUIVALENT, 0 ms] (148) TRUE (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, 2003 ms] (201) QDP (202) QDPOrderProof [EQUIVALENT, 2131 ms] (203) QDP (204) QDPOrderProof [EQUIVALENT, 1854 ms] (205) QDP (206) QDPOrderProof [EQUIVALENT, 300 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. ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2) 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) 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) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = x_1 POL(ISNAT(x_1)) = x_1 POL(ISNATILIST(x_1)) = x_1 POL(ISNATILISTKIND(x_1)) = x_1 POL(ISNATKIND(x_1)) = x_1 POL(ISNATLIST(x_1)) = x_1 POL(LENGTH(x_1)) = x_1 POL(TAKE(x_1, x_2)) = x_1 + x_2 POL(U101(x_1, x_2, x_3)) = 0 POL(U101^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U102(x_1, x_2, x_3)) = 0 POL(U102^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U103(x_1, x_2, x_3)) = 0 POL(U103^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U104(x_1, x_2, x_3)) = 0 POL(U104^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U105(x_1, x_2)) = 0 POL(U105^1(x_1, x_2)) = x_2 POL(U106(x_1)) = 0 POL(U11(x_1, x_2)) = 0 POL(U111(x_1, x_2, x_3)) = x_2 POL(U111^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U112(x_1, x_2, x_3)) = x_2 POL(U112^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U113(x_1, x_2, x_3)) = x_2 POL(U113^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U114(x_1, x_2)) = x_2 POL(U114^1(x_1, x_2)) = x_2 POL(U11^1(x_1, x_2)) = x_2 POL(U12(x_1, x_2)) = 0 POL(U121(x_1, x_2)) = x_2 POL(U121^1(x_1, x_2)) = x_2 POL(U122(x_1)) = 0 POL(U12^1(x_1, x_2)) = x_2 POL(U13(x_1)) = 0 POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(U131^1(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4 POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(U132^1(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4 POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(U133^1(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4 POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(U134^1(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4 POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(U135^1(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4 POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(U136^1(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4 POL(U21(x_1, x_2)) = 0 POL(U21^1(x_1, x_2)) = x_2 POL(U22(x_1, x_2)) = 0 POL(U22^1(x_1, x_2)) = x_2 POL(U23(x_1)) = 0 POL(U31(x_1, x_2)) = 0 POL(U31^1(x_1, x_2)) = x_2 POL(U32(x_1, x_2)) = 0 POL(U32^1(x_1, x_2)) = x_2 POL(U33(x_1)) = 0 POL(U41(x_1, x_2, x_3)) = 0 POL(U41^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U42(x_1, x_2, x_3)) = 0 POL(U42^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U43(x_1, x_2, x_3)) = 0 POL(U43^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U44(x_1, x_2, x_3)) = 0 POL(U44^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U45(x_1, x_2)) = 0 POL(U45^1(x_1, x_2)) = x_2 POL(U46(x_1)) = 0 POL(U51(x_1, x_2)) = 0 POL(U51^1(x_1, x_2)) = x_2 POL(U52(x_1)) = 0 POL(U61(x_1, x_2)) = 0 POL(U61^1(x_1, x_2)) = x_2 POL(U62(x_1)) = 0 POL(U71(x_1)) = 0 POL(U81(x_1)) = 0 POL(U91(x_1, x_2, x_3)) = 0 POL(U91^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U92(x_1, x_2, x_3)) = 0 POL(U92^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U93(x_1, x_2, x_3)) = 0 POL(U93^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U94(x_1, x_2, x_3)) = 0 POL(U94^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U95(x_1, x_2)) = 0 POL(U95^1(x_1, x_2)) = x_2 POL(U96(x_1)) = 0 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatIListKind(x_1)) = 0 POL(isNatKind(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = x_1 + x_2 POL(n__length(x_1)) = x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__take(x_1, x_2)) = 1 + x_1 + x_2 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 1 + x_1 + x_2 POL(tt) = 0 POL(zeros) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 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__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 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__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 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: 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) 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)) U32^1(tt, V) -> ACTIVATE(V) U31^1(tt, V) -> ISNATILISTKIND(activate(V)) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2) U61^1(tt, V2) -> ISNATILISTKIND(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. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 79 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) 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)) U51^1(tt, V2) -> ACTIVATE(V2) 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)) -> ISNATILISTKIND(activate(V1)) ISNATILISTKIND(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) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) U95^1(tt, V2) -> ACTIVATE(V2) U94^1(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) 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) 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. ACTIVATE(n__length(X)) -> LENGTH(X) ISNATKIND(n__length(V1)) -> ISNATILISTKIND(activate(V1)) ISNATKIND(n__length(V1)) -> ACTIVATE(V1) ISNAT(n__length(V1)) -> U11^1(isNatIListKind(activate(V1)), activate(V1)) ISNAT(n__length(V1)) -> ISNATILISTKIND(activate(V1)) ISNAT(n__length(V1)) -> ACTIVATE(V1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = x_1 POL(ISNAT(x_1)) = x_1 POL(ISNATILISTKIND(x_1)) = x_1 POL(ISNATKIND(x_1)) = x_1 POL(ISNATLIST(x_1)) = x_1 POL(LENGTH(x_1)) = x_1 POL(U101(x_1, x_2, x_3)) = 0 POL(U102(x_1, x_2, x_3)) = 0 POL(U103(x_1, x_2, x_3)) = 0 POL(U104(x_1, x_2, x_3)) = 0 POL(U105(x_1, x_2)) = 0 POL(U106(x_1)) = 0 POL(U11(x_1, x_2)) = 0 POL(U111(x_1, x_2, x_3)) = 1 + x_2 POL(U111^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U112(x_1, x_2, x_3)) = 1 + x_2 POL(U112^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U113(x_1, x_2, x_3)) = 1 + x_2 POL(U113^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U114(x_1, x_2)) = 1 + x_2 POL(U114^1(x_1, x_2)) = x_2 POL(U11^1(x_1, x_2)) = x_2 POL(U12(x_1, x_2)) = 0 POL(U121(x_1, x_2)) = x_2 POL(U122(x_1)) = 0 POL(U12^1(x_1, x_2)) = x_2 POL(U13(x_1)) = 0 POL(U131(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(U132(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(U133(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(U134(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(U135(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(U136(x_1, x_2, x_3, x_4)) = x_2 + x_4 POL(U21(x_1, x_2)) = 0 POL(U21^1(x_1, x_2)) = x_2 POL(U22(x_1, x_2)) = 0 POL(U22^1(x_1, x_2)) = x_2 POL(U23(x_1)) = 0 POL(U31(x_1, x_2)) = 0 POL(U32(x_1, x_2)) = 0 POL(U33(x_1)) = 0 POL(U41(x_1, x_2, x_3)) = 0 POL(U42(x_1, x_2, x_3)) = 0 POL(U43(x_1, x_2, x_3)) = 0 POL(U44(x_1, x_2, x_3)) = 0 POL(U45(x_1, x_2)) = 0 POL(U46(x_1)) = 0 POL(U51(x_1, x_2)) = 0 POL(U51^1(x_1, x_2)) = x_2 POL(U52(x_1)) = 0 POL(U61(x_1, x_2)) = 0 POL(U62(x_1)) = 0 POL(U71(x_1)) = 0 POL(U81(x_1)) = 0 POL(U91(x_1, x_2, x_3)) = 0 POL(U91^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U92(x_1, x_2, x_3)) = 0 POL(U92^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U93(x_1, x_2, x_3)) = 0 POL(U93^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U94(x_1, x_2, x_3)) = 0 POL(U94^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U95(x_1, x_2)) = 0 POL(U95^1(x_1, x_2)) = x_2 POL(U96(x_1)) = 0 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatIListKind(x_1)) = 0 POL(isNatKind(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 1 + x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = x_1 + x_2 POL(n__length(x_1)) = 1 + x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__take(x_1, x_2)) = x_2 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = x_2 POL(tt) = 0 POL(zeros) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 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__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 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) 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: U51^1(tt, V2) -> ISNATILISTKIND(activate(V2)) ISNATILISTKIND(n__cons(V1, V2)) -> U51^1(isNatKind(activate(V1)), activate(V2)) U51^1(tt, V2) -> ACTIVATE(V2) 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)) ISNATILISTKIND(n__cons(V1, V2)) -> ISNATKIND(activate(V1)) ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1)) ISNATKIND(n__s(V1)) -> ACTIVATE(V1) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) U95^1(tt, V2) -> ACTIVATE(V2) U94^1(tt, V1, V2) -> ISNAT(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__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) 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) 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 5 SCCs with 43 less nodes. ---------------------------------------- (13) Complex Obligation (AND) ---------------------------------------- (14) 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. ---------------------------------------- (15) 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)) = [[2A]] + [[0A]] * 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)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * 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 + [[0A]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(tt) = [[2A]] >>> <<< POL(U31(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNatIListKind(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U111(x_1, x_2, x_3)) = [[-I]] + [[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)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U71(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U51(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U81(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U52(x_1)) = [[2A]] + [[-I]] * 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)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U13(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U106(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * 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)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[3A]] + [[-I]] * 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)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U33(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U122(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[2A]] + [[-I]] * x_1 + [[1A]] * 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 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__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 ---------------------------------------- (16) 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. ---------------------------------------- (17) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (18) YES ---------------------------------------- (19) 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. ---------------------------------------- (20) 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)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(n__s(x_1)) = [[2A]] + [[1A]] * x_1 >>> <<< POL(U21^1(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNatKind(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(activate(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(tt) = [[2A]] >>> <<< POL(U22^1(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__zeros) = [[0A]] >>> <<< POL(zeros) = [[1A]] >>> <<< POL(n__take(x_1, x_2)) = [[2A]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[2A]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__0) = [[0A]] >>> <<< POL(0) = [[0A]] >>> <<< POL(n__length(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(length(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[2A]] + [[1A]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(cons(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__nil) = [[2A]] >>> <<< POL(nil) = [[2A]] >>> <<< POL(U71(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(isNatIListKind(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U81(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U121(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U111(x_1, x_2, x_3)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(isNatList(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U51(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U52(x_1)) = [[2A]] + [[-I]] * 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 + [[0A]] * 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)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U13(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U106(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * 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)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[3A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U114(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U32(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U33(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U122(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[-I]] + [[1A]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[2A]] * x_4 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[2A]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[-I]] + [[1A]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[2A]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[2A]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[1A]] * 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)) 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 ---------------------------------------- (21) 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. ---------------------------------------- (22) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (23) TRUE ---------------------------------------- (24) 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, V2) -> ISNATILISTKIND(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. ---------------------------------------- (25) 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))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, V2) -> ISNATILISTKIND(activate(V2)) 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. ---------------------------------------- (27) 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)) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: 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(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. ---------------------------------------- (29) 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))) ---------------------------------------- (30) 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)) 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(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, 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. ---------------------------------------- (31) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__zeros) -> ISNATILISTKIND(zeros) 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)) 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. ---------------------------------------- (33) 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)) ---------------------------------------- (34) 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)) 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. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (36) 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)) 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. ---------------------------------------- (37) 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)) ---------------------------------------- (38) 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)) 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)) 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. ---------------------------------------- (39) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (40) 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__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)) 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. ---------------------------------------- (41) 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))) ---------------------------------------- (42) 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__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. ---------------------------------------- (43) 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))) ---------------------------------------- (44) 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__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. ---------------------------------------- (45) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (46) 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__cons(x0, x1)) -> ISNATILISTKIND(cons(x0, x1)) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(x0)), activate(y1)) U51^1(tt, n__nil) -> ISNATILISTKIND(nil) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1)) U51^1(tt, x0) -> ISNATILISTKIND(x0) ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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)) 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. ---------------------------------------- (47) 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))) ---------------------------------------- (48) 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__nil) -> ISNATILISTKIND(nil) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1)) U51^1(tt, x0) -> ISNATILISTKIND(x0) ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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)) 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)) 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. ---------------------------------------- (49) 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))) ---------------------------------------- (50) 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__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1)) U51^1(tt, x0) -> ISNATILISTKIND(x0) ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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)) 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)) 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. ---------------------------------------- (51) 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)) ---------------------------------------- (52) 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__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(x0, x1)), activate(y1)) U51^1(tt, x0) -> ISNATILISTKIND(x0) ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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)) 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__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. ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (54) 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, x0) -> ISNATILISTKIND(x0) ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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__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. ---------------------------------------- (55) 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))) ---------------------------------------- (56) 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(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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__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)) 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. ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (58) 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(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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__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. ---------------------------------------- (59) 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))) ---------------------------------------- (60) 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) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) 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__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)) 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. ---------------------------------------- (61) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (62) 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__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__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1)) 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. ---------------------------------------- (63) 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))) ---------------------------------------- (64) 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) 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__cons(x0, x1)) -> ISNATILISTKIND(n__cons(x0, x1)) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(n__s(x0)), activate(y1)) 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. ---------------------------------------- (65) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (66) 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__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)) 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. ---------------------------------------- (67) 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))) ---------------------------------------- (68) 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__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)) ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(n__0, n__zeros)), activate(y0)) 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. ---------------------------------------- (69) 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))) ---------------------------------------- (70) 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__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. ---------------------------------------- (71) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (72) 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)) 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. ---------------------------------------- (73) 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))) ---------------------------------------- (74) 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(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. ---------------------------------------- (75) 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 POL( U51^1_2(x_1, x_2) ) = 2x_2 POL( take_2(x_1, x_2) ) = 0 POL( 0 ) = 0 POL( U121_2(x_1, x_2) ) = max{0, -2} POL( isNatIList_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2x_2 POL( U131_4(x_1, ..., x_4) ) = max{0, -2} POL( activate_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = 0 POL( U101_3(x_1, ..., x_3) ) = max{0, -2} POL( U102_3(x_1, ..., x_3) ) = max{0, -2} POL( U114_2(x_1, x_2) ) = max{0, -2} POL( U134_4(x_1, ..., x_4) ) = max{0, -2} POL( U136_4(x_1, ..., x_4) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = max{0, -2} POL( U42_3(x_1, ..., x_3) ) = 0 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = max{0, -2} POL( U92_3(x_1, ..., x_3) ) = max{0, -2} POL( isNatKind_1(x_1) ) = max{0, -2} POL( n__0 ) = 0 POL( tt ) = 0 POL( n__length_1(x_1) ) = 2 POL( U71_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = max{0, -2} POL( n__s_1(x_1) ) = 0 POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U105_2(x_1, x_2) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = max{0, -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, -2} POL( U12_2(x_1, x_2) ) = max{0, -2} POL( U132_4(x_1, ..., x_4) ) = max{0, -2} POL( U133_4(x_1, ..., x_4) ) = max{0, -2} POL( U135_4(x_1, ..., x_4) ) = max{0, -2} POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = max{0, -2} POL( U43_3(x_1, ..., x_3) ) = max{0, -2} POL( U44_3(x_1, ..., x_3) ) = max{0, -2} POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U93_3(x_1, ..., x_3) ) = 0 POL( U94_3(x_1, ..., x_3) ) = max{0, -2} POL( U95_2(x_1, x_2) ) = max{0, -2} POL( isNat_1(x_1) ) = max{0, -2} POL( U106_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = max{0, -2} POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__cons_2(x_1, x_2) ) = 2x_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__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(V) -> U31(isNatIListKind(activate(V)), activate(V)) 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__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 ---------------------------------------- (76) 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)) 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(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. ---------------------------------------- (77) 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)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNATILISTKIND_1(x_1) ) = 2x_1 POL( U51^1_2(x_1, x_2) ) = 2x_2 POL( take_2(x_1, x_2) ) = x_2 + 2 POL( 0 ) = 0 POL( U121_2(x_1, x_2) ) = 2 POL( isNatIList_1(x_1) ) = 1 POL( s_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( U131_4(x_1, ..., x_4) ) = x_2 + x_4 + 2 POL( activate_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_2 + 2 POL( U101_3(x_1, ..., x_3) ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U114_2(x_1, x_2) ) = max{0, -2} POL( U134_4(x_1, ..., x_4) ) = x_2 + x_4 + 2 POL( U136_4(x_1, ..., x_4) ) = x_2 + x_4 + 2 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = 1 POL( U42_3(x_1, ..., x_3) ) = 1 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U81_1(x_1) ) = 0 POL( U91_3(x_1, ..., x_3) ) = 2 POL( U92_3(x_1, ..., x_3) ) = 2 POL( isNatKind_1(x_1) ) = max{0, -2} POL( n__0 ) = 0 POL( tt ) = 0 POL( n__length_1(x_1) ) = 0 POL( U71_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = 0 POL( n__s_1(x_1) ) = 0 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 1 POL( U105_2(x_1, x_2) ) = 1 POL( U11_2(x_1, x_2) ) = max{0, -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, -2} POL( U12_2(x_1, x_2) ) = 0 POL( U132_4(x_1, ..., x_4) ) = x_2 + x_4 + 2 POL( U133_4(x_1, ..., x_4) ) = x_2 + x_4 + 2 POL( U135_4(x_1, ..., x_4) ) = x_2 + x_4 + 2 POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = max{0, -2} POL( U43_3(x_1, ..., x_3) ) = 1 POL( U44_3(x_1, ..., x_3) ) = 1 POL( U45_2(x_1, x_2) ) = max{0, -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, -2} POL( isNat_1(x_1) ) = max{0, -2} POL( U106_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = 0 POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = max{0, -2} POL( n__zeros ) = 0 POL( zeros ) = 0 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__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(V) -> U31(isNatIListKind(activate(V)), activate(V)) 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__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 ---------------------------------------- (78) 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__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1)) 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(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. ---------------------------------------- (79) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(x0)), activate(y1)) 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) ) = x_1 POL( U51^1_2(x_1, x_2) ) = x_2 POL( take_2(x_1, x_2) ) = x_2 POL( 0 ) = 0 POL( U121_2(x_1, x_2) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( activate_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_2 POL( length_1(x_1) ) = 2 POL( nil ) = 0 POL( U111_3(x_1, ..., x_3) ) = 2 POL( isNatList_1(x_1) ) = max{0, -2} POL( n__length_1(x_1) ) = 2 POL( U101_3(x_1, ..., x_3) ) = max{0, -2} POL( U102_3(x_1, ..., x_3) ) = 0 POL( U114_2(x_1, x_2) ) = 2 POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U136_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = max{0, -2} POL( U42_3(x_1, ..., x_3) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = max{0, -2} POL( U92_3(x_1, ..., x_3) ) = max{0, -2} POL( isNatKind_1(x_1) ) = max{0, -2} POL( n__0 ) = 0 POL( tt ) = 0 POL( U71_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = max{0, -2} POL( n__s_1(x_1) ) = 2 POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U105_2(x_1, x_2) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = 2 POL( U112_3(x_1, ..., x_3) ) = 2 POL( U113_3(x_1, ..., x_3) ) = 2 POL( U12_2(x_1, x_2) ) = max{0, -2} POL( U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U135_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = max{0, -2} POL( U43_3(x_1, ..., x_3) ) = max{0, -2} POL( U44_3(x_1, ..., x_3) ) = max{0, -2} POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U93_3(x_1, ..., x_3) ) = max{0, -2} POL( U94_3(x_1, ..., x_3) ) = max{0, -2} POL( U95_2(x_1, x_2) ) = max{0, -2} POL( isNat_1(x_1) ) = 2x_1 + 2 POL( U106_1(x_1) ) = 0 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = 0 POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = 0 POL( U62_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_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__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)) 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__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 ---------------------------------------- (80) 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(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)) 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. ---------------------------------------- (81) 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)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(tt) = [[2A]] >>> <<< POL(n__take(x_1, x_2)) = [[2A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(ISNATILISTKIND(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(take(x_1, x_2)) = [[2A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__cons(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNatKind(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(activate(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(n__0) = [[0A]] >>> <<< POL(n__zeros) = [[0A]] >>> <<< POL(0) = [[0A]] >>> <<< POL(U121(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(s(x_1)) = [[2A]] + [[1A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[3A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 + [[0A]] * 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)) = [[2A]] + [[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) = [[2A]] >>> <<< POL(nil) = [[2A]] >>> <<< POL(U31(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U111(x_1, x_2, x_3)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(isNatList(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U51(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U52(x_1)) = [[2A]] + [[-I]] * 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 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U93(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U94(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U95(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U13(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U23(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U106(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[2A]] + [[-I]] * 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)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * 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(U132(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[3A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 + [[0A]] * 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)) 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__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 ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: 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)) 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. ---------------------------------------- (83) 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. ---------------------------------------- (84) NO ---------------------------------------- (85) 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. ---------------------------------------- (86) 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))) ---------------------------------------- (87) 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. ---------------------------------------- (88) 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)) ---------------------------------------- (89) 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. ---------------------------------------- (90) 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))) ---------------------------------------- (91) 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. ---------------------------------------- (92) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (93) 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. ---------------------------------------- (94) 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)) ---------------------------------------- (95) 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. ---------------------------------------- (96) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (97) 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. ---------------------------------------- (98) 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)) ---------------------------------------- (99) 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. ---------------------------------------- (100) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (101) 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. ---------------------------------------- (102) 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))) ---------------------------------------- (103) 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. ---------------------------------------- (104) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (105) 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. ---------------------------------------- (106) 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))) ---------------------------------------- (107) 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. ---------------------------------------- (108) 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)) ---------------------------------------- (109) 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. ---------------------------------------- (110) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (111) 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. ---------------------------------------- (112) 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))) ---------------------------------------- (113) 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. ---------------------------------------- (114) 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))) ---------------------------------------- (115) 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. ---------------------------------------- (116) 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))) ---------------------------------------- (117) 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. ---------------------------------------- (118) 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))) ---------------------------------------- (119) 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. ---------------------------------------- (120) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (121) 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. ---------------------------------------- (122) 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))) ---------------------------------------- (123) 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. ---------------------------------------- (124) 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))) ---------------------------------------- (125) 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. ---------------------------------------- (126) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (127) 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. ---------------------------------------- (128) 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))) ---------------------------------------- (129) 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. ---------------------------------------- (130) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (131) 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. ---------------------------------------- (132) 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))) ---------------------------------------- (133) 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. ---------------------------------------- (134) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (135) 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. ---------------------------------------- (136) 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 + 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( ISNATLIST_1(x_1) ) = x_1 POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2x_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) ) = 1 POL( U111_3(x_1, ..., x_3) ) = max{0, -2} POL( U112_3(x_1, ..., x_3) ) = max{0, -2} POL( U113_3(x_1, ..., x_3) ) = max{0, -2} POL( U114_2(x_1, x_2) ) = max{0, -2} POL( U12_2(x_1, x_2) ) = max{0, -2} POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 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) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 1 POL( U45_2(x_1, x_2) ) = max{0, -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, -2} POL( U92_3(x_1, ..., x_3) ) = max{0, -2} POL( U93_3(x_1, ..., x_3) ) = max{0, -2} POL( U94_3(x_1, ..., x_3) ) = max{0, -2} POL( U95_2(x_1, x_2) ) = max{0, -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) ) = 2 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = 0 POL( U46_1(x_1) ) = 0 POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U71_1(x_1) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( length_1(x_1) ) = 0 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) ) = 2x_1 + 2x_2 + 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( n__s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 0 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 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) 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(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 ---------------------------------------- (137) 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. ---------------------------------------- (138) 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)) 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 + 2x_2 + 2x_3 + 2 POL( U92^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U93^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U94^1_3(x_1, ..., x_3) ) = 2x_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, -2} POL( U102_3(x_1, ..., x_3) ) = max{0, -2} POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U105_2(x_1, x_2) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = max{0, -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, -2} POL( U114_2(x_1, x_2) ) = max{0, -2} POL( U12_2(x_1, x_2) ) = max{0, -2} POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U135_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U136_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = max{0, -2} POL( U42_3(x_1, ..., x_3) ) = max{0, -2} POL( U43_3(x_1, ..., x_3) ) = max{0, -2} POL( U44_3(x_1, ..., x_3) ) = max{0, -2} POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = 1 POL( U92_3(x_1, ..., x_3) ) = 1 POL( U93_3(x_1, ..., x_3) ) = 1 POL( U94_3(x_1, ..., x_3) ) = 1 POL( U95_2(x_1, x_2) ) = 1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatKind_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatIListKind_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = max{0, -2} POL( U106_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U71_1(x_1) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__take_2(x_1, x_2) ) = x_2 POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = max{0, -2} POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( take_2(x_1, x_2) ) = x_2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 1 POL( n__s_1(x_1) ) = 0 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 0 POL( U121_2(x_1, x_2) ) = max{0, -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__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__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)) 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 ---------------------------------------- (139) 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)) 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. ---------------------------------------- (140) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 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) ) = 2x_2 + 2x_3 + 2 POL( U92^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U93^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U94^1_3(x_1, ..., x_3) ) = 2x_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) ) = 1 POL( U102_3(x_1, ..., x_3) ) = 1 POL( U103_3(x_1, ..., x_3) ) = 1 POL( U104_3(x_1, ..., x_3) ) = 1 POL( U105_2(x_1, x_2) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = max{0, -2} POL( U111_3(x_1, ..., x_3) ) = 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) ) = 0 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2 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) ) = 2x_2 + 2x_4 + 2 POL( U136_4(x_1, ..., x_4) ) = 2x_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) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = 1 POL( U42_3(x_1, ..., x_3) ) = 1 POL( U43_3(x_1, ..., x_3) ) = 1 POL( U44_3(x_1, ..., x_3) ) = 1 POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = max{0, -2} POL( U92_3(x_1, ..., x_3) ) = max{0, -2} POL( U93_3(x_1, ..., x_3) ) = max{0, -2} POL( U94_3(x_1, ..., x_3) ) = max{0, -2} POL( U95_2(x_1, x_2) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatKind_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 2x_1 POL( U106_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 1 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = 0 POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U71_1(x_1) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__take_2(x_1, x_2) ) = 2x_2 + 2 POL( length_1(x_1) ) = 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 + 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 ) = 0 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__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) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) 0 -> n__0 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)) 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 ---------------------------------------- (141) 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. ---------------------------------------- (142) 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 + 2 POL( U92^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U93^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U94^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U95^1_2(x_1, x_2) ) = 2x_2 + 2 POL( ISNATLIST_1(x_1) ) = x_1 + 2 POL( U101_3(x_1, ..., x_3) ) = 1 POL( U102_3(x_1, ..., x_3) ) = 1 POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U105_2(x_1, x_2) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = max{0, -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) ) = max{0, -2} POL( U131_4(x_1, ..., x_4) ) = max{0, -2} POL( U132_4(x_1, ..., x_4) ) = max{0, -2} POL( U133_4(x_1, ..., x_4) ) = max{0, -2} POL( U134_4(x_1, ..., x_4) ) = max{0, -2} POL( U135_4(x_1, ..., x_4) ) = max{0, -2} POL( U136_4(x_1, ..., x_4) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = 2 POL( U92_3(x_1, ..., x_3) ) = 2 POL( U93_3(x_1, ..., x_3) ) = 2 POL( U94_3(x_1, ..., x_3) ) = 2 POL( U95_2(x_1, x_2) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2x_2 POL( isNatKind_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = max{0, -2} POL( U106_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = 0 POL( U71_1(x_1) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__take_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 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 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 POL( n__s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 0 POL( U121_2(x_1, x_2) ) = max{0, -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__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) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) 0 -> n__0 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)) 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 ---------------------------------------- (143) 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. ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: 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)) 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. U112^1(tt, L, N) -> U113^1(isNat(activate(N)), activate(L), activate(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 POL( U111^1_3(x_1, ..., x_3) ) = x_1 + 2x_2 POL( U112^1_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U113^1_3(x_1, ..., x_3) ) = 2x_2 POL( U114^1_2(x_1, x_2) ) = 2x_2 POL( U101_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U102_3(x_1, ..., x_3) ) = 2x_2 + x_3 POL( U103_3(x_1, ..., x_3) ) = 2x_2 POL( U104_3(x_1, ..., x_3) ) = 2x_2 POL( U105_2(x_1, x_2) ) = x_1 POL( U11_2(x_1, x_2) ) = 2x_2 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) ) = 2x_2 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 POL( U21_2(x_1, x_2) ) = 2x_2 POL( U22_2(x_1, x_2) ) = 2x_2 POL( U31_2(x_1, x_2) ) = 1 POL( U32_2(x_1, x_2) ) = 1 POL( U41_3(x_1, ..., x_3) ) = 1 POL( U42_3(x_1, ..., x_3) ) = 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) ) = 2 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) ) = 2x_3 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) ) = 2 POL( isNat_1(x_1) ) = 2x_1 POL( isNatKind_1(x_1) ) = 1 POL( U106_1(x_1) ) = max{0, 2x_1 - 1} POL( isNatIList_1(x_1) ) = 1 POL( U122_1(x_1) ) = 1 POL( U23_1(x_1) ) = x_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( n__take_2(x_1, x_2) ) = x_1 + x_2 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 + x_2 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 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__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__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)) 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 ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: 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)) 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. ---------------------------------------- (147) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. ---------------------------------------- (148) TRUE ---------------------------------------- (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__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 POL( U42^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 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) ) = x_1 + 2x_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) ) = 1 POL( U11_2(x_1, x_2) ) = max{0, -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, -2} POL( U114_2(x_1, x_2) ) = max{0, -2} POL( U12_2(x_1, x_2) ) = max{0, -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) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = 0 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 1 POL( U41_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 1 POL( U42_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U43_3(x_1, ..., x_3) ) = 0 POL( U44_3(x_1, ..., x_3) ) = max{0, -2} POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 1 POL( U61_2(x_1, x_2) ) = 2 POL( U91_3(x_1, ..., x_3) ) = max{0, -2} POL( U92_3(x_1, ..., x_3) ) = 0 POL( U93_3(x_1, ..., x_3) ) = max{0, -2} POL( U94_3(x_1, ..., x_3) ) = max{0, -2} POL( U95_2(x_1, x_2) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatKind_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = 2 POL( isNat_1(x_1) ) = max{0, 2x_1 - 1} POL( U106_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = x_1 + 2 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = 1 POL( U62_1(x_1) ) = 1 POL( U71_1(x_1) ) = 0 POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__take_2(x_1, x_2) ) = 2x_2 POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = max{0, -2} POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( take_2(x_1, x_2) ) = 2x_2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 1 POL( n__s_1(x_1) ) = 0 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 0 POL( U121_2(x_1, x_2) ) = max{0, -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__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__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(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 ---------------------------------------- (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, 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. ---------------------------------------- (202) 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 POL( U42^1_3(x_1, ..., x_3) ) = 2x_3 POL( U43^1_3(x_1, ..., x_3) ) = 2x_3 POL( U44^1_3(x_1, ..., x_3) ) = 2x_3 POL( U45^1_2(x_1, x_2) ) = 2x_2 POL( ISNATILIST_1(x_1) ) = x_1 POL( U101_3(x_1, ..., x_3) ) = max{0, -2} POL( U102_3(x_1, ..., x_3) ) = max{0, -2} POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U105_2(x_1, x_2) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = 2 POL( U111_3(x_1, ..., x_3) ) = 2 POL( U112_3(x_1, ..., x_3) ) = 2 POL( U113_3(x_1, ..., x_3) ) = 2 POL( U114_2(x_1, x_2) ) = 2 POL( U12_2(x_1, x_2) ) = 2 POL( U131_4(x_1, ..., x_4) ) = max{0, -2} POL( U132_4(x_1, ..., x_4) ) = max{0, -2} POL( U133_4(x_1, ..., x_4) ) = max{0, -2} POL( U134_4(x_1, ..., x_4) ) = max{0, -2} POL( U135_4(x_1, ..., x_4) ) = max{0, -2} POL( U136_4(x_1, ..., x_4) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = max{0, -2} POL( U42_3(x_1, ..., x_3) ) = max{0, -2} POL( U43_3(x_1, ..., x_3) ) = 0 POL( U44_3(x_1, ..., x_3) ) = max{0, -2} POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = 1 POL( U92_3(x_1, ..., x_3) ) = 0 POL( U93_3(x_1, ..., x_3) ) = max{0, -2} POL( U94_3(x_1, ..., x_3) ) = max{0, -2} POL( U95_2(x_1, x_2) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2x_2 POL( isNatKind_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 2x_1 + 2 POL( U106_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = 0 POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U71_1(x_1) ) = 0 POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__take_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 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 ) = 2 POL( 0 ) = 2 POL( n__length_1(x_1) ) = 2 POL( n__s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 0 POL( U121_2(x_1, x_2) ) = max{0, -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__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__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 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(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, 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)) 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. ---------------------------------------- (204) 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)) 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) ) = x_2 + 2x_3 + 1 POL( U42^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 1 POL( U43^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 1 POL( U44^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 1 POL( U45^1_2(x_1, x_2) ) = 2x_2 + 1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 1 POL( U101_3(x_1, ..., x_3) ) = 2 POL( U102_3(x_1, ..., x_3) ) = 0 POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U105_2(x_1, x_2) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = max{0, -2} POL( U111_3(x_1, ..., x_3) ) = 2 POL( U112_3(x_1, ..., x_3) ) = 2 POL( U113_3(x_1, ..., x_3) ) = 1 POL( U114_2(x_1, x_2) ) = 1 POL( U12_2(x_1, x_2) ) = max{0, -2} POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( U135_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( U136_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_2(x_1, x_2) ) = 1 POL( U32_2(x_1, x_2) ) = 1 POL( U41_3(x_1, ..., x_3) ) = max{0, -2} POL( U42_3(x_1, ..., x_3) ) = max{0, -2} POL( U43_3(x_1, ..., x_3) ) = max{0, -2} POL( U44_3(x_1, ..., x_3) ) = max{0, -2} POL( U45_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = x_2 + 2 POL( U92_3(x_1, ..., x_3) ) = 1 POL( U93_3(x_1, ..., x_3) ) = 1 POL( U94_3(x_1, ..., x_3) ) = 1 POL( U95_2(x_1, x_2) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatKind_1(x_1) ) = max{0, -2} POL( isNatIListKind_1(x_1) ) = 2 POL( isNat_1(x_1) ) = max{0, -2} POL( U106_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = x_1 + 2 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = max{0, -2} POL( U46_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U71_1(x_1) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = max{0, -2} POL( n__take_2(x_1, x_2) ) = x_2 + 1 POL( length_1(x_1) ) = 2 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 + 1 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 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 ) = 0 POL( U121_2(x_1, x_2) ) = max{0, -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__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__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 0 -> n__0 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__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)) 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. U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(x0, x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(U43^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 >>> <<< POL(tt) = [[2A]] >>> <<< POL(U44^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 >>> <<< POL(isNatIListKind(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(activate(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U45^1(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[2A]] * x_2 >>> <<< POL(n__take(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(ISNATILIST(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(take(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__cons(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U41^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 >>> <<< POL(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U42^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 >>> <<< POL(n__zeros) = [[0A]] >>> <<< POL(0) = [[1A]] >>> <<< POL(n__0) = [[1A]] >>> <<< POL(isNat(x_1)) = [[-I]] + [[1A]] * x_1 >>> <<< POL(zeros) = [[1A]] >>> <<< POL(n__length(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(length(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(n__s(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__nil) = [[2A]] >>> <<< POL(nil) = [[2A]] >>> <<< POL(U51(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U121(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[3A]] + [[2A]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[3A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[1A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U71(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U81(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U31(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[2A]] * x_2 >>> <<< POL(U111(x_1, x_2, x_3)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(isNatList(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U52(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U92(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U93(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U94(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U95(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U13(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U23(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U106(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U45(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U46(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U96(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * 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 + [[1A]] * x_2 >>> <<< POL(U33(x_1)) = [[-I]] + [[1A]] * x_1 >>> <<< POL(U122(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[-I]] + [[1A]] * x_1 + [[2A]] * x_2 + [[1A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[3A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[1A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[1A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[1A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[1A]] * x_3 + [[0A]] * 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 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) isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 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)) 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)) 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.