NO proof of Transformed_CSR_04_OvConsOS_complete-noand_FR.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 58 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 3718 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 2461 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 3563 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) AND (17) QDP (18) QDPOrderProof [EQUIVALENT, 700 ms] (19) QDP (20) DependencyGraphProof [EQUIVALENT, 0 ms] (21) TRUE (22) QDP (23) QDPOrderProof [EQUIVALENT, 304 ms] (24) QDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) TransformationProof [EQUIVALENT, 0 ms] (32) QDP (33) DependencyGraphProof [EQUIVALENT, 0 ms] (34) QDP (35) TransformationProof [EQUIVALENT, 0 ms] (36) QDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) QDP (39) TransformationProof [EQUIVALENT, 0 ms] (40) QDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 0 ms] (44) QDP (45) TransformationProof [EQUIVALENT, 0 ms] (46) QDP (47) DependencyGraphProof [EQUIVALENT, 0 ms] (48) QDP (49) TransformationProof [EQUIVALENT, 16 ms] (50) QDP (51) DependencyGraphProof [EQUIVALENT, 0 ms] (52) QDP (53) TransformationProof [EQUIVALENT, 0 ms] (54) QDP (55) DependencyGraphProof [EQUIVALENT, 0 ms] (56) QDP (57) TransformationProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) DependencyGraphProof [EQUIVALENT, 0 ms] (62) QDP (63) TransformationProof [EQUIVALENT, 8 ms] (64) QDP (65) QDPOrderProof [EQUIVALENT, 269 ms] (66) QDP (67) QDPOrderProof [EQUIVALENT, 278 ms] (68) QDP (69) QDPOrderProof [EQUIVALENT, 442 ms] (70) QDP (71) QDPOrderProof [EQUIVALENT, 426 ms] (72) QDP (73) QDPOrderProof [EQUIVALENT, 1387 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 186 ms] (76) QDP (77) NonTerminationLoopProof [COMPLETE, 257 ms] (78) NO (79) QDP (80) TransformationProof [EQUIVALENT, 0 ms] (81) QDP (82) TransformationProof [EQUIVALENT, 0 ms] (83) QDP (84) TransformationProof [EQUIVALENT, 0 ms] (85) QDP (86) DependencyGraphProof [EQUIVALENT, 0 ms] (87) QDP (88) TransformationProof [EQUIVALENT, 0 ms] (89) QDP (90) DependencyGraphProof [EQUIVALENT, 0 ms] (91) QDP (92) TransformationProof [EQUIVALENT, 0 ms] (93) QDP (94) DependencyGraphProof [EQUIVALENT, 0 ms] (95) QDP (96) TransformationProof [EQUIVALENT, 0 ms] (97) QDP (98) DependencyGraphProof [EQUIVALENT, 0 ms] (99) QDP (100) TransformationProof [EQUIVALENT, 0 ms] (101) QDP (102) TransformationProof [EQUIVALENT, 0 ms] (103) QDP (104) TransformationProof [EQUIVALENT, 0 ms] (105) QDP (106) DependencyGraphProof [EQUIVALENT, 0 ms] (107) QDP (108) TransformationProof [EQUIVALENT, 0 ms] (109) QDP (110) TransformationProof [EQUIVALENT, 0 ms] (111) QDP (112) DependencyGraphProof [EQUIVALENT, 0 ms] (113) QDP (114) TransformationProof [EQUIVALENT, 0 ms] (115) QDP (116) DependencyGraphProof [EQUIVALENT, 0 ms] (117) QDP (118) QDPOrderProof [EQUIVALENT, 758 ms] (119) QDP (120) QDPOrderProof [EQUIVALENT, 248 ms] (121) QDP (122) QDPOrderProof [EQUIVALENT, 609 ms] (123) QDP (124) QDPOrderProof [EQUIVALENT, 772 ms] (125) QDP (126) QDPOrderProof [EQUIVALENT, 1526 ms] (127) QDP (128) QDP (129) TransformationProof [EQUIVALENT, 0 ms] (130) QDP (131) TransformationProof [EQUIVALENT, 0 ms] (132) QDP (133) TransformationProof [EQUIVALENT, 0 ms] (134) QDP (135) DependencyGraphProof [EQUIVALENT, 0 ms] (136) QDP (137) TransformationProof [EQUIVALENT, 0 ms] (138) QDP (139) DependencyGraphProof [EQUIVALENT, 0 ms] (140) QDP (141) TransformationProof [EQUIVALENT, 0 ms] (142) QDP (143) DependencyGraphProof [EQUIVALENT, 0 ms] (144) QDP (145) TransformationProof [EQUIVALENT, 0 ms] (146) QDP (147) DependencyGraphProof [EQUIVALENT, 0 ms] (148) QDP (149) TransformationProof [EQUIVALENT, 0 ms] (150) QDP (151) TransformationProof [EQUIVALENT, 4 ms] (152) QDP (153) TransformationProof [EQUIVALENT, 0 ms] (154) QDP (155) DependencyGraphProof [EQUIVALENT, 0 ms] (156) QDP (157) TransformationProof [EQUIVALENT, 0 ms] (158) QDP (159) TransformationProof [EQUIVALENT, 0 ms] (160) QDP (161) DependencyGraphProof [EQUIVALENT, 0 ms] (162) QDP (163) TransformationProof [EQUIVALENT, 0 ms] (164) QDP (165) DependencyGraphProof [EQUIVALENT, 0 ms] (166) QDP (167) QDPOrderProof [EQUIVALENT, 626 ms] (168) QDP (169) QDPOrderProof [EQUIVALENT, 591 ms] (170) QDP (171) QDPOrderProof [EQUIVALENT, 706 ms] (172) QDP (173) QDPOrderProof [EQUIVALENT, 781 ms] (174) QDP (175) QDPOrderProof [EQUIVALENT, 1715 ms] (176) QDP (177) QDP (178) QDPOrderProof [EQUIVALENT, 513 ms] (179) QDP (180) DependencyGraphProof [EQUIVALENT, 0 ms] (181) TRUE (182) QDP (183) QDPOrderProof [EQUIVALENT, 570 ms] (184) QDP (185) DependencyGraphProof [EQUIVALENT, 0 ms] (186) AND (187) QDP (188) UsableRulesProof [EQUIVALENT, 0 ms] (189) QDP (190) QDPSizeChangeProof [EQUIVALENT, 1 ms] (191) YES (192) QDP (193) QDPOrderProof [EQUIVALENT, 603 ms] (194) QDP (195) PisEmptyProof [EQUIVALENT, 0 ms] (196) YES ---------------------------------------- (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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), activate(X2)) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__0) -> 0^1 ACTIVATE(n__length(X)) -> LENGTH(activate(X)) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> S(activate(X)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> CONS(activate(X1), X2) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), activate(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__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__length(X)) -> LENGTH(activate(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) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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. U31^1(tt, V) -> U32^1(isNatIListKind(activate(V)), activate(V)) U31^1(tt, V) -> ISNATILISTKIND(activate(V)) U31^1(tt, V) -> ACTIVATE(V) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNAT_1(x_1) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 2x_1 POL( ISNATILISTKIND_1(x_1) ) = x_1 POL( ISNATKIND_1(x_1) ) = 2x_1 POL( ISNATLIST_1(x_1) ) = x_1 POL( LENGTH_1(x_1) ) = x_1 POL( TAKE_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( U101^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U102^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U103^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U104^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U105^1_2(x_1, x_2) ) = 2x_2 POL( U111^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( U112^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( U113^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( U114^1_2(x_1, x_2) ) = x_2 POL( U11^1_2(x_1, x_2) ) = 2x_2 POL( U12^1_2(x_1, x_2) ) = 2x_2 POL( U131^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U132^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U133^1_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + 2x_4 - 1} POL( U134^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U135^1_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U136^1_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U21^1_2(x_1, x_2) ) = 2x_2 POL( U22^1_2(x_1, x_2) ) = 2x_2 POL( U31^1_2(x_1, x_2) ) = x_1 + x_2 POL( U32^1_2(x_1, x_2) ) = x_2 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) ) = 2x_2 POL( U51^1_2(x_1, x_2) ) = x_2 POL( U61^1_2(x_1, x_2) ) = 2x_2 POL( U91^1_3(x_1, ..., x_3) ) = 2x_2 + x_3 POL( U92^1_3(x_1, ..., x_3) ) = 2x_2 + x_3 POL( U93^1_3(x_1, ..., x_3) ) = 2x_2 + x_3 POL( U94^1_3(x_1, ..., x_3) ) = 2x_2 + x_3 POL( U95^1_2(x_1, x_2) ) = x_2 POL( U101_3(x_1, ..., x_3) ) = 1 POL( U121^1_2(x_1, x_2) ) = max{0, x_1 + x_2 - 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) ) = x_1 POL( U11_2(x_1, x_2) ) = 1 POL( U111_3(x_1, ..., x_3) ) = 2x_2 POL( U112_3(x_1, ..., x_3) ) = 2x_2 POL( U113_3(x_1, ..., x_3) ) = 2x_2 POL( U114_2(x_1, x_2) ) = 2x_2 POL( U12_2(x_1, x_2) ) = 1 POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 + 2x_4 - 2} POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = 1 POL( U31_2(x_1, x_2) ) = 1 POL( U32_2(x_1, x_2) ) = 1 POL( U41_3(x_1, ..., x_3) ) = 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) ) = x_2 POL( U61_2(x_1, x_2) ) = 2x_2 POL( U91_3(x_1, ..., x_3) ) = 2x_3 POL( U92_3(x_1, ..., x_3) ) = 2x_3 POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U94_3(x_1, ..., x_3) ) = 1 POL( U95_2(x_1, x_2) ) = max{0, 2x_1 - 1} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatKind_1(x_1) ) = 2x_1 + 2 POL( length_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( isNatIListKind_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = 2x_1 + 1 POL( isNatIList_1(x_1) ) = 1 POL( U106_1(x_1) ) = max{0, 2x_1 - 1} POL( U122_1(x_1) ) = 2x_1 POL( U13_1(x_1) ) = 1 POL( U23_1(x_1) ) = 1 POL( U33_1(x_1) ) = 1 POL( U46_1(x_1) ) = max{0, 2x_1 - 1} POL( U52_1(x_1) ) = x_1 POL( U62_1(x_1) ) = max{0, 2x_1 - 1} POL( U71_1(x_1) ) = max{0, 2x_1 - 1} POL( U96_1(x_1) ) = 1 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( U81_1(x_1) ) = 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 POL( n__s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 2 POL( nil ) = 2 POL( U121_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1} POL( tt ) = 1 POL( ACTIVATE_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X 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)) 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__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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 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 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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(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__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__length(X)) -> LENGTH(activate(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) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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)) 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) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2)) U61^1(tt, V2) -> ISNATILISTKIND(activate(V2)) ISNATILISTKIND(n__take(V1, V2)) -> ISNATKIND(activate(V1)) ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V1) ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V2) U61^1(tt, V2) -> ACTIVATE(V2) ISNATILIST(V) -> 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 1 SCC with 3 less nodes. ---------------------------------------- (8) 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__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) TAKE(0, IL) -> ISNATILIST(IL) ISNATILIST(V) -> ISNATILISTKIND(activate(V)) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__length(X)) -> LENGTH(activate(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) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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) -> 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)) ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2) U12^1(tt, V1) -> ACTIVATE(V1) U11^1(tt, V1) -> ISNATILISTKIND(activate(V1)) 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) 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) TAKE(s(M), cons(N, IL)) -> U131^1(isNatIList(activate(IL)), activate(IL), M, N) U131^1(tt, IL, M, N) -> U132^1(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132^1(tt, IL, M, N) -> U133^1(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133^1(tt, IL, M, N) -> U134^1(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135^1(tt, IL, M, N) -> U136^1(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136^1(tt, IL, M, N) -> ACTIVATE(N) U136^1(tt, IL, M, N) -> ACTIVATE(M) U136^1(tt, IL, M, N) -> ACTIVATE(IL) U135^1(tt, IL, M, N) -> ISNATKIND(activate(N)) U135^1(tt, IL, M, N) -> ACTIVATE(N) U135^1(tt, IL, M, N) -> ACTIVATE(IL) U135^1(tt, IL, M, N) -> ACTIVATE(M) U134^1(tt, IL, M, N) -> ISNAT(activate(N)) U134^1(tt, IL, M, N) -> ACTIVATE(N) U134^1(tt, IL, M, N) -> ACTIVATE(IL) U134^1(tt, IL, M, N) -> ACTIVATE(M) U133^1(tt, IL, M, N) -> ISNATKIND(activate(M)) U133^1(tt, IL, M, N) -> ACTIVATE(M) U133^1(tt, IL, M, N) -> ACTIVATE(IL) U133^1(tt, IL, M, N) -> ACTIVATE(N) U132^1(tt, IL, M, N) -> ISNAT(activate(M)) U132^1(tt, IL, M, N) -> ACTIVATE(M) U132^1(tt, IL, M, N) -> ACTIVATE(IL) U132^1(tt, IL, M, N) -> ACTIVATE(N) U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL)) U131^1(tt, IL, M, N) -> ACTIVATE(IL) U131^1(tt, IL, M, N) -> ACTIVATE(M) U131^1(tt, IL, M, N) -> ACTIVATE(N) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL) U51^1(tt, V2) -> ACTIVATE(V2) U121^1(tt, IL) -> ACTIVATE(IL) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U11^1(tt, V1) -> U12^1(isNatIListKind(activate(V1)), activate(V1)) U11^1(tt, V1) -> ISNATILISTKIND(activate(V1)) U11^1(tt, V1) -> ACTIVATE(V1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNAT_1(x_1) ) = 2x_1 + 2 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( ISNATILISTKIND_1(x_1) ) = 2x_1 + 2 POL( ISNATKIND_1(x_1) ) = 2x_1 + 2 POL( ISNATLIST_1(x_1) ) = 2x_1 + 2 POL( LENGTH_1(x_1) ) = 2x_1 + 2 POL( TAKE_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U101^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U102^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U103^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U104^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U105^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U111^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( U112^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U113^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U114^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U11^1_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( U121^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U12^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U131^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U132^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U133^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U134^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U135^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U136^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U21^1_2(x_1, x_2) ) = x_1 + 2x_2 POL( U22^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U41^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( U42^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( U43^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U44^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U45^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U51^1_2(x_1, x_2) ) = x_1 + 2x_2 POL( U61^1_2(x_1, x_2) ) = 2x_2 + 2 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( isNatIList_1(x_1) ) = 2 POL( U31_2(x_1, x_2) ) = 2 POL( isNatIListKind_1(x_1) ) = 2x_1 + 1 POL( activate_1(x_1) ) = x_1 POL( U106_1(x_1) ) = 2 POL( U121_2(x_1, x_2) ) = 2x_2 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U46_1(x_1) ) = 2 POL( n__zeros ) = 1 POL( tt ) = 2 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( isNatKind_1(x_1) ) = 2 POL( U101_3(x_1, ..., x_3) ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2 POL( U105_2(x_1, x_2) ) = 2 POL( U11_2(x_1, x_2) ) = x_1 POL( U111_3(x_1, ..., x_3) ) = 2x_2 POL( U112_3(x_1, ..., x_3) ) = 2x_2 POL( U113_3(x_1, ..., x_3) ) = 2x_2 POL( U114_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U12_2(x_1, x_2) ) = 2 POL( U132_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U133_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 POL( U21_2(x_1, x_2) ) = x_1 POL( U22_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2x_2 + 1 POL( U61_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1} POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U93_3(x_1, ..., x_3) ) = 2 POL( U94_3(x_1, ..., x_3) ) = 2 POL( U95_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( length_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = x_1 + 2x_2 POL( isNat_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = max{0, x_1 - 1} POL( U13_1(x_1) ) = x_1 POL( U23_1(x_1) ) = 2 POL( U33_1(x_1) ) = x_1 POL( U52_1(x_1) ) = x_1 POL( U62_1(x_1) ) = x_1 POL( U71_1(x_1) ) = 2 POL( U96_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_1 + 2x_2 POL( U81_1(x_1) ) = 2 POL( zeros ) = 1 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 POL( n__s_1(x_1) ) = x_1 POL( n__nil ) = 1 POL( nil ) = 1 POL( ACTIVATE_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNat(n__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) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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 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 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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (10) 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__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) TAKE(0, IL) -> ISNATILIST(IL) ISNATILIST(V) -> ISNATILISTKIND(activate(V)) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__length(X)) -> LENGTH(activate(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) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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) -> ACTIVATE(V) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U41^1(tt, V1, V2) -> U42^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U42^1(tt, V1, V2) -> U43^1(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43^1(tt, V1, V2) -> U44^1(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44^1(tt, V1, V2) -> U45^1(isNat(activate(V1)), activate(V2)) U45^1(tt, V2) -> ISNATILIST(activate(V2)) ISNATILIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) U45^1(tt, V2) -> ACTIVATE(V2) U44^1(tt, V1, V2) -> ISNAT(activate(V1)) ISNAT(n__length(V1)) -> U11^1(isNatIListKind(activate(V1)), activate(V1)) U12^1(tt, V1) -> ISNATLIST(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2) U12^1(tt, V1) -> ACTIVATE(V1) 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) 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) TAKE(s(M), cons(N, IL)) -> U131^1(isNatIList(activate(IL)), activate(IL), M, N) U131^1(tt, IL, M, N) -> U132^1(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132^1(tt, IL, M, N) -> U133^1(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133^1(tt, IL, M, N) -> U134^1(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135^1(tt, IL, M, N) -> U136^1(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136^1(tt, IL, M, N) -> ACTIVATE(N) U136^1(tt, IL, M, N) -> ACTIVATE(M) U136^1(tt, IL, M, N) -> ACTIVATE(IL) U135^1(tt, IL, M, N) -> ISNATKIND(activate(N)) U135^1(tt, IL, M, N) -> ACTIVATE(N) U135^1(tt, IL, M, N) -> ACTIVATE(IL) U135^1(tt, IL, M, N) -> ACTIVATE(M) U134^1(tt, IL, M, N) -> ISNAT(activate(N)) U134^1(tt, IL, M, N) -> ACTIVATE(N) U134^1(tt, IL, M, N) -> ACTIVATE(IL) U134^1(tt, IL, M, N) -> ACTIVATE(M) U133^1(tt, IL, M, N) -> ISNATKIND(activate(M)) U133^1(tt, IL, M, N) -> ACTIVATE(M) U133^1(tt, IL, M, N) -> ACTIVATE(IL) U133^1(tt, IL, M, N) -> ACTIVATE(N) U132^1(tt, IL, M, N) -> ISNAT(activate(M)) U132^1(tt, IL, M, N) -> ACTIVATE(M) U132^1(tt, IL, M, N) -> ACTIVATE(IL) U132^1(tt, IL, M, N) -> ACTIVATE(N) U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL)) U131^1(tt, IL, M, N) -> ACTIVATE(IL) U131^1(tt, IL, M, N) -> ACTIVATE(M) U131^1(tt, IL, M, N) -> ACTIVATE(N) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL) U51^1(tt, V2) -> ACTIVATE(V2) U121^1(tt, IL) -> ACTIVATE(IL) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 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__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) TAKE(0, IL) -> U121^1(isNatIList(IL), IL) U121^1(tt, IL) -> ACTIVATE(IL) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__length(X)) -> LENGTH(activate(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) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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) -> 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) 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)) -> 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) ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2) U95^1(tt, V2) -> ACTIVATE(V2) U94^1(tt, V1, V2) -> ISNAT(activate(V1)) U94^1(tt, V1, V2) -> ACTIVATE(V1) U94^1(tt, V1, V2) -> ACTIVATE(V2) U93^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U93^1(tt, V1, V2) -> ACTIVATE(V2) U93^1(tt, V1, V2) -> ACTIVATE(V1) U92^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U92^1(tt, V1, V2) -> ACTIVATE(V2) U92^1(tt, V1, V2) -> ACTIVATE(V1) U91^1(tt, V1, V2) -> ISNATKIND(activate(V1)) U91^1(tt, V1, V2) -> ACTIVATE(V1) U91^1(tt, V1, V2) -> ACTIVATE(V2) 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) 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) U51^1(tt, V2) -> 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILISTKIND(n__cons(V1, V2)) -> ISNATKIND(activate(V1)) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V1) U121^1(tt, IL) -> ACTIVATE(IL) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ISNATLIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1)) ISNATKIND(n__length(V1)) -> ACTIVATE(V1) ACTIVATE(n__length(X)) -> ACTIVATE(X) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATILISTKIND(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATILISTKIND(n__take(V1, V2)) -> ISNATKIND(activate(V1)) ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V1) ISNATILISTKIND(n__take(V1, V2)) -> ACTIVATE(V2) U61^1(tt, V2) -> ACTIVATE(V2) ISNATILIST(V) -> ACTIVATE(V) ISNATILIST(n__cons(V1, V2)) -> ISNATKIND(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) U45^1(tt, V2) -> ACTIVATE(V2) ISNAT(n__length(V1)) -> ISNATILISTKIND(activate(V1)) ISNAT(n__length(V1)) -> 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) -> ACTIVATE(V2) U43^1(tt, V1, V2) -> ACTIVATE(V1) 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) -> ACTIVATE(V1) U104^1(tt, V1, V2) -> ACTIVATE(V2) U103^1(tt, V1, V2) -> ACTIVATE(V2) U103^1(tt, V1, V2) -> ACTIVATE(V1) U102^1(tt, V1, V2) -> ACTIVATE(V2) U102^1(tt, V1, V2) -> ACTIVATE(V1) U101^1(tt, V1, V2) -> ISNATKIND(activate(V1)) U101^1(tt, V1, V2) -> ACTIVATE(V1) U101^1(tt, V1, V2) -> ACTIVATE(V2) ISNATLIST(n__take(V1, V2)) -> ISNATKIND(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2) U95^1(tt, V2) -> ACTIVATE(V2) U94^1(tt, V1, V2) -> ACTIVATE(V1) U94^1(tt, V1, V2) -> ACTIVATE(V2) U93^1(tt, V1, V2) -> ACTIVATE(V2) U93^1(tt, V1, V2) -> ACTIVATE(V1) 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) -> ACTIVATE(N) U112^1(tt, L, N) -> ACTIVATE(L) U111^1(tt, L, N) -> ACTIVATE(L) U111^1(tt, L, N) -> 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) -> ACTIVATE(M) U132^1(tt, IL, M, N) -> ACTIVATE(IL) U132^1(tt, IL, M, N) -> ACTIVATE(N) 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)) -> ACTIVATE(IL) U51^1(tt, V2) -> ACTIVATE(V2) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNAT_1(x_1) ) = x_1 + 2 POL( ISNATILIST_1(x_1) ) = x_1 + 2 POL( ISNATILISTKIND_1(x_1) ) = x_1 + 2 POL( ISNATKIND_1(x_1) ) = x_1 POL( ISNATLIST_1(x_1) ) = 2x_1 + 2 POL( LENGTH_1(x_1) ) = 2x_1 + 2 POL( TAKE_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U101^1_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2 POL( U102^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U103^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U104^1_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 POL( U105^1_2(x_1, x_2) ) = x_2 + 2 POL( U111^1_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 + 2 POL( U112^1_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2 POL( U113^1_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2 POL( U114^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U131^1_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 2 POL( U132^1_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + x_4 + 2 POL( U133^1_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2 POL( U134^1_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2 POL( U135^1_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + x_4 POL( U136^1_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 POL( U21^1_2(x_1, x_2) ) = x_2 + 2 POL( U22^1_2(x_1, x_2) ) = x_2 + 2 POL( U41^1_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2 POL( U42^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U43^1_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 POL( U44^1_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U45^1_2(x_1, x_2) ) = x_2 + 2 POL( U51^1_2(x_1, x_2) ) = x_2 + 2 POL( U61^1_2(x_1, x_2) ) = x_1 + x_2 + 2 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) ) = x_1 + 2x_2 + 2x_3 + 2 POL( U94^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U95^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U101_3(x_1, ..., x_3) ) = 0 POL( U121^1_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( U102_3(x_1, ..., x_3) ) = 0 POL( U103_3(x_1, ..., x_3) ) = x_1 POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U105_2(x_1, x_2) ) = 0 POL( U11_2(x_1, x_2) ) = 1 POL( U111_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U112_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U113_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U114_2(x_1, x_2) ) = 2x_2 + 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) ) = 1 POL( U22_2(x_1, x_2) ) = 1 POL( U31_2(x_1, x_2) ) = 0 POL( U32_2(x_1, x_2) ) = max{0, -1} POL( U41_3(x_1, ..., x_3) ) = max{0, -2} POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U43_3(x_1, ..., x_3) ) = max{0, -1} POL( U44_3(x_1, ..., x_3) ) = 0 POL( U45_2(x_1, x_2) ) = 0 POL( U51_2(x_1, x_2) ) = 0 POL( U61_2(x_1, x_2) ) = 0 POL( U91_3(x_1, ..., x_3) ) = 0 POL( U92_3(x_1, ..., x_3) ) = 0 POL( U93_3(x_1, ..., x_3) ) = max{0, -1} POL( U94_3(x_1, ..., x_3) ) = max{0, -1} POL( U95_2(x_1, x_2) ) = max{0, x_1 - 1} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatKind_1(x_1) ) = 2 POL( length_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( isNatIListKind_1(x_1) ) = max{0, -1} POL( isNat_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( U106_1(x_1) ) = 0 POL( U122_1(x_1) ) = x_1 + 1 POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = 1 POL( U33_1(x_1) ) = 0 POL( U46_1(x_1) ) = max{0, -1} POL( U52_1(x_1) ) = x_1 POL( U62_1(x_1) ) = x_1 POL( U71_1(x_1) ) = 1 POL( U96_1(x_1) ) = 0 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U81_1(x_1) ) = 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( n__s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 1 POL( nil ) = 1 POL( tt ) = 0 POL( U121_2(x_1, x_2) ) = x_2 + 2 POL( ACTIVATE_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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))) 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)) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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 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 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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 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)) ISNATKIND(n__length(V1)) -> ISNATILISTKIND(activate(V1)) ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) TAKE(0, IL) -> U121^1(isNatIList(IL), IL) ACTIVATE(n__length(X)) -> LENGTH(activate(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)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1)) ISNATKIND(n__s(V1)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> U101^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U101^1(tt, V1, V2) -> U102^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U102^1(tt, V1, V2) -> U103^1(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103^1(tt, V1, V2) -> U104^1(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104^1(tt, V1, V2) -> U105^1(isNat(activate(V1)), activate(V2)) U105^1(tt, V2) -> ISNATILIST(activate(V2)) ISNATILIST(V) -> ISNATILISTKIND(activate(V)) ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2)) U61^1(tt, V2) -> ISNATILISTKIND(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)) 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)) U44^1(tt, V1, V2) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> U21^1(isNatKind(activate(V1)), activate(V1)) U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1)) U22^1(tt, V1) -> ISNAT(activate(V1)) U43^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U42^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U104^1(tt, V1, V2) -> ISNAT(activate(V1)) U103^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U102^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U94^1(tt, V1, V2) -> ISNAT(activate(V1)) U93^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U92^1(tt, V1, V2) -> ISNATILISTKIND(activate(V2)) U112^1(tt, L, N) -> ISNAT(activate(N)) U111^1(tt, L, N) -> ISNATILISTKIND(activate(L)) 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)) U132^1(tt, IL, M, N) -> ISNAT(activate(M)) U131^1(tt, IL, M, N) -> ISNATILISTKIND(activate(IL)) TAKE(s(M), cons(N, IL)) -> ISNATILIST(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 28 less nodes. ---------------------------------------- (16) Complex Obligation (AND) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1)) U22^1(tt, V1) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> U21^1(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) 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(U21^1(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[3A]] * x_2 >>> <<< POL(tt) = [[5A]] >>> <<< POL(U22^1(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[3A]] * x_2 >>> <<< POL(isNatKind(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(activate(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(ISNAT(x_1)) = [[2A]] + [[2A]] * x_1 >>> <<< POL(n__s(x_1)) = [[5A]] + [[1A]] * x_1 >>> <<< POL(n__zeros) = [[1A]] >>> <<< POL(zeros) = [[2A]] >>> <<< POL(n__take(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__0) = [[5A]] >>> <<< POL(0) = [[5A]] >>> <<< POL(n__length(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(length(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[5A]] + [[1A]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(cons(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__nil) = [[5A]] >>> <<< POL(nil) = [[5A]] >>> <<< POL(U71(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(isNatIListKind(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U81(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U111(x_1, x_2, x_3)) = [[3A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(isNatList(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U51(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U52(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U62(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U92(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U93(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U94(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U95(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNat(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[4A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[4A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U13(x_1)) = [[4A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U23(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U106(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[5A]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U32(x_1, x_2)) = [[5A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U33(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U45(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U46(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U96(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[5A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U114(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[2A]] * x_2 >>> <<< POL(U121(x_1, x_2)) = [[4A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U122(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[5A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[4A]] + [[-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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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))) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(tt, V1) -> U22^1(isNatKind(activate(V1)), activate(V1)) ISNAT(n__s(V1)) -> U21^1(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (21) TRUE ---------------------------------------- (22) 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)) ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2)) U61^1(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILISTKIND(n__take(V1, V2)) -> U61^1(isNatKind(activate(V1)), activate(V2)) 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( U61^1_2(x_1, x_2) ) = 2x_2 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 1} POL( U102_3(x_1, ..., x_3) ) = 2x_2 POL( U103_3(x_1, ..., x_3) ) = 2x_2 POL( U104_3(x_1, ..., x_3) ) = 2x_2 POL( U105_2(x_1, x_2) ) = x_1 POL( U11_2(x_1, x_2) ) = 1 POL( U111_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U112_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U113_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U114_2(x_1, x_2) ) = 2x_2 + 2 POL( U12_2(x_1, x_2) ) = 1 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2 POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2 POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2 POL( U21_2(x_1, x_2) ) = x_2 POL( U22_2(x_1, x_2) ) = x_2 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 1 POL( U45_2(x_1, x_2) ) = 1 POL( U51_2(x_1, x_2) ) = x_2 + 1 POL( U61_2(x_1, x_2) ) = 2x_1 + 1 POL( U91_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U92_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U94_3(x_1, ..., x_3) ) = 1 POL( U95_2(x_1, x_2) ) = 1 POL( cons_2(x_1, x_2) ) = x_2 POL( isNatKind_1(x_1) ) = x_1 + 1 POL( length_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( isNatIListKind_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = x_1 POL( U106_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2x_1 + 1 POL( U122_1(x_1) ) = x_1 + 2 POL( U13_1(x_1) ) = 1 POL( U23_1(x_1) ) = x_1 POL( U33_1(x_1) ) = 2 POL( U46_1(x_1) ) = 1 POL( U52_1(x_1) ) = x_1 POL( U62_1(x_1) ) = 1 POL( U71_1(x_1) ) = 1 POL( U96_1(x_1) ) = 1 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 1 POL( 0 ) = 1 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( n__s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = x_2 POL( n__nil ) = 1 POL( nil ) = 1 POL( U81_1(x_1) ) = 1 POL( tt ) = 1 POL( U121_2(x_1, x_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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (24) 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)) U61^1(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (26) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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(activate(x0), activate(x1))), activate(y1)),ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(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(activate(x0))), activate(y1)),ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1))) (ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)),ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1))) (ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)),ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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))) ---------------------------------------- (28) 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(activate(x0), activate(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(activate(x0))), activate(y1)) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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(activate(x0), activate(x1))),U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1)))) (U51^1(tt, n__0) -> ISNATILISTKIND(0),U51^1(tt, n__0) -> ISNATILISTKIND(0)) (U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))),U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0)))) (U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))),U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0)))) (U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)),U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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)) ---------------------------------------- (30) 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(activate(x0), activate(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(activate(x0))), activate(y1)) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(x0), activate(x1))) U51^1(tt, n__0) -> ISNATILISTKIND(0) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) 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))) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(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(activate(x0))), activate(y1)) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(x0), activate(x1))) U51^1(tt, n__0) -> ISNATILISTKIND(0) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (34) 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(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(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(activate(x0))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) 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)) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(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(activate(x0))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(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(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) 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)) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (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(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__0, y1)) -> U51^1(isNatKind(0), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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))) ---------------------------------------- (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(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) 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)) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) ISNATILISTKIND(n__cons(n__nil, y1)) -> U51^1(isNatKind(nil), activate(y1)) U51^1(tt, x0) -> ISNATILISTKIND(x0) ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0)) ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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__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))) ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), 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)) 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(x0, y1)) -> U51^1(isNatKind(x0), activate(y1)) U51^1(tt, x0) -> ISNATILISTKIND(x0) ISNATILISTKIND(n__cons(n__zeros, y0)) -> U51^1(isNatKind(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__zeros) -> ISNATILISTKIND(cons(0, n__zeros)) ISNATILISTKIND(n__cons(n__0, y0)) -> U51^1(isNatKind(n__0), activate(y0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) 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))) ---------------------------------------- (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(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) 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))) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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__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))) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) -> U51^1(isNatKind(take(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(x0), x1)) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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))) ---------------------------------------- (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(activate(x0), activate(x1))), activate(y1)) U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) 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(activate(x0), activate(x1))), activate(y1)) ISNATILISTKIND(n__cons(n__length(x0), y1)) -> U51^1(isNatKind(length(activate(x0))), activate(y1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U51^1_2(x_1, x_2) ) = x_2 + 2 POL( ISNATILISTKIND_1(x_1) ) = x_1 + 2 POL( U101_3(x_1, ..., x_3) ) = 1 POL( U102_3(x_1, ..., x_3) ) = x_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) ) = 0 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) ) = max{0, -2} 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) ) = x_1 + 2x_2 + x_4 + 2 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2 POL( U21_2(x_1, x_2) ) = 0 POL( U22_2(x_1, x_2) ) = 0 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = x_2 + 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = x_1 POL( U61_2(x_1, x_2) ) = x_1 POL( U91_3(x_1, ..., x_3) ) = x_1 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, -1} POL( U95_2(x_1, x_2) ) = max{0, -1} POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, -2} POL( take_2(x_1, x_2) ) = 2x_2 + 2 POL( isNatKind_1(x_1) ) = 1 POL( isNatIListKind_1(x_1) ) = 1 POL( isNat_1(x_1) ) = x_1 POL( U106_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = 1 POL( U122_1(x_1) ) = 2 POL( U13_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( U33_1(x_1) ) = 2 POL( U46_1(x_1) ) = 2 POL( U52_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = 0 POL( U71_1(x_1) ) = max{0, 2x_1 - 2} POL( U81_1(x_1) ) = 0 POL( U96_1(x_1) ) = 0 POL( n__take_2(x_1, x_2) ) = 2x_2 + 2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 POL( n__s_1(x_1) ) = 0 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( n__nil ) = 1 POL( nil ) = 1 POL( U121_2(x_1, x_2) ) = 2x_2 + 2 POL( tt ) = 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X 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))) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILISTKIND(n__cons(n__s(x0), y1)) -> U51^1(isNatKind(s(activate(x0))), activate(y1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U51^1_2(x_1, x_2) ) = x_2 + 2 POL( ISNATILISTKIND_1(x_1) ) = x_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) ) = 0 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) ) = 0 POL( U111_3(x_1, ..., x_3) ) = 2 POL( U112_3(x_1, ..., x_3) ) = 2 POL( U113_3(x_1, ..., x_3) ) = x_1 + 2 POL( U114_2(x_1, x_2) ) = 2 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 + x_4 + 2 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2 POL( U21_2(x_1, x_2) ) = 0 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) ) = 1 POL( U42_3(x_1, ..., x_3) ) = max{0, -1} 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) ) = 0 POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U93_3(x_1, ..., x_3) ) = max{0, -1} POL( U94_3(x_1, ..., x_3) ) = 0 POL( U95_2(x_1, x_2) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( length_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = 1 POL( take_2(x_1, x_2) ) = 2x_2 + 2 POL( isNatKind_1(x_1) ) = 2 POL( isNatIListKind_1(x_1) ) = max{0, -1} POL( isNat_1(x_1) ) = max{0, -1} POL( U106_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = max{0, -2} POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = x_1 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) ) = 2x_1 POL( U71_1(x_1) ) = 0 POL( U81_1(x_1) ) = 0 POL( U96_1(x_1) ) = 0 POL( n__take_2(x_1, x_2) ) = 2x_2 + 2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 1 POL( zeros ) = 1 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( n__s_1(x_1) ) = 1 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U121_2(x_1, x_2) ) = 2x_2 POL( tt ) = 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X 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) s(X) -> n__s(X) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) U51^1(tt, n__length(x0)) -> ISNATILISTKIND(length(activate(x0))) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) 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(activate(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U51^1_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( ISNATILISTKIND_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) ) = 1 POL( U104_3(x_1, ..., x_3) ) = 1 POL( U105_2(x_1, x_2) ) = 1 POL( U11_2(x_1, x_2) ) = 2 POL( U111_3(x_1, ..., x_3) ) = 0 POL( U112_3(x_1, ..., x_3) ) = 0 POL( U113_3(x_1, ..., x_3) ) = max{0, -2} POL( U114_2(x_1, x_2) ) = 0 POL( U12_2(x_1, x_2) ) = 2 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 1 POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = x_1 POL( U51_2(x_1, x_2) ) = x_1 POL( U61_2(x_1, x_2) ) = x_2 + 2 POL( U91_3(x_1, ..., x_3) ) = 2 POL( U92_3(x_1, ..., x_3) ) = 2 POL( U93_3(x_1, ..., x_3) ) = 2 POL( U94_3(x_1, ..., x_3) ) = 2 POL( U95_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = max{0, -2} POL( take_2(x_1, x_2) ) = x_2 POL( isNatKind_1(x_1) ) = 1 POL( isNatIListKind_1(x_1) ) = x_1 + 2 POL( isNat_1(x_1) ) = 2 POL( U106_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = 0 POL( U13_1(x_1) ) = 2 POL( U23_1(x_1) ) = 1 POL( U33_1(x_1) ) = 1 POL( U46_1(x_1) ) = 1 POL( U52_1(x_1) ) = 1 POL( U62_1(x_1) ) = x_1 POL( U71_1(x_1) ) = 1 POL( U81_1(x_1) ) = 1 POL( U96_1(x_1) ) = 1 POL( n__take_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 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 + 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2} POL( tt ) = 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X 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) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 0 -> n__0 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U51^1(tt, n__s(x0)) -> ISNATILISTKIND(s(activate(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U51^1_2(x_1, x_2) ) = 2x_2 + 1 POL( ISNATILISTKIND_1(x_1) ) = x_1 + 1 POL( U101_3(x_1, ..., x_3) ) = x_1 + 2x_3 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U104_3(x_1, ..., x_3) ) = 2 POL( U105_2(x_1, x_2) ) = 2 POL( U11_2(x_1, x_2) ) = 2 POL( U111_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 1} POL( U112_3(x_1, ..., x_3) ) = 1 POL( U113_3(x_1, ..., x_3) ) = max{0, x_1 - 1} POL( U114_2(x_1, x_2) ) = max{0, x_1 - 1} POL( U12_2(x_1, x_2) ) = 2 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U132_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_4 - 2} POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_4 - 2} POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U21_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U22_2(x_1, x_2) ) = 2 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = x_1 POL( U61_2(x_1, x_2) ) = 2 POL( U91_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} POL( U93_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U94_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U95_2(x_1, x_2) ) = x_1 + 2x_2 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( length_1(x_1) ) = 2x_1 + 1 POL( s_1(x_1) ) = 1 POL( take_2(x_1, x_2) ) = 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) ) = 2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( U122_1(x_1) ) = 0 POL( U13_1(x_1) ) = 2 POL( U23_1(x_1) ) = x_1 POL( U33_1(x_1) ) = 2 POL( U46_1(x_1) ) = x_1 POL( U52_1(x_1) ) = 2 POL( U62_1(x_1) ) = 2 POL( U71_1(x_1) ) = x_1 POL( U81_1(x_1) ) = 2 POL( U96_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 + 1 POL( n__s_1(x_1) ) = 1 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U121_2(x_1, x_2) ) = 0 POL( tt ) = 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X 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) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 0 -> n__0 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)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__take(x0, x1)) -> ISNATILISTKIND(take(activate(x0), activate(x1))) ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) 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(activate(x0), activate(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)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 >>> <<< POL(tt) = [[2A]] >>> <<< POL(n__take(x_1, x_2)) = [[3A]] + [[2A]] * x_1 + [[1A]] * x_2 >>> <<< POL(ISNATILISTKIND(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(take(x_1, x_2)) = [[3A]] + [[2A]] * x_1 + [[1A]] * x_2 >>> <<< POL(activate(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNatKind(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__0) = [[2A]] >>> <<< POL(n__zeros) = [[0A]] >>> <<< POL(0) = [[2A]] >>> <<< 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(n__nil) = [[2A]] >>> <<< POL(nil) = [[2A]] >>> <<< POL(U121(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[2A]] + [[2A]] * x_1 + [[2A]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U71(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(isNatIListKind(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U81(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U111(x_1, x_2, x_3)) = [[1A]] + [[1A]] * x_1 + [[1A]] * x_2 + [[-I]] * x_3 >>> <<< POL(isNatList(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U51(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U52(x_1)) = [[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)) = [[1A]] + [[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)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[2A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U23(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U106(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U32(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U33(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U45(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U46(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U96(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[3A]] + [[-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(U122(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[-I]] + [[2A]] * x_1 + [[2A]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[4A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[3A]] + [[2A]] * x_1 + [[2A]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[4A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[4A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X 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) cons(X1, X2) -> n__cons(X1, X2) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 0 -> n__0 length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(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)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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. ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNatKind(cons(activate(x0), x1)), activate(y1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(ISNATILISTKIND(x_1)) = [[1]] + [[0, 1]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[1], [0]] + [[0, 1], [1, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 >>> <<< POL(U51^1(x_1, x_2)) = [[1]] + [[0, 0]] * x_1 + [[0, 1]] * x_2 >>> <<< POL(isNatKind(x_1)) = [[1], [0]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[1], [0]] + [[0, 1], [1, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 >>> <<< POL(activate(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(tt) = [[1], [0]] >>> <<< POL(n__0) = [[0], [0]] >>> <<< POL(n__zeros) = [[1], [1]] >>> <<< POL(0) = [[0], [0]] >>> <<< POL(zeros) = [[1], [1]] >>> <<< POL(n__take(x_1, x_2)) = [[1], [1]] + [[0, 1], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[1], [1]] + [[0, 1], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__length(x_1)) = [[1], [1]] + [[0, 0], [0, 1]] * x_1 >>> <<< POL(length(x_1)) = [[1], [1]] + [[0, 0], [0, 1]] * x_1 >>> <<< POL(n__s(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(s(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__nil) = [[0], [0]] >>> <<< POL(nil) = [[0], [0]] >>> <<< POL(U71(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(isNatIListKind(x_1)) = [[1], [1]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U81(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U111(x_1, x_2, x_3)) = [[1], [1]] + [[0, 0], [0, 1]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 >>> <<< POL(isNatList(x_1)) = [[1], [0]] + [[1, 1], [0, 1]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[0, 1], [0, 0]] * x_3 >>> <<< POL(U51(x_1, x_2)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U52(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U62(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U92(x_1, x_2, x_3)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 1], [0, 0]] * x_3 >>> <<< POL(U93(x_1, x_2, x_3)) = [[1], [0]] + [[0, 1], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 1], [0, 0]] * x_3 >>> <<< POL(U94(x_1, x_2, x_3)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 1], [0, 0]] * x_3 >>> <<< POL(U95(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 1], [0, 0]] * x_2 >>> <<< POL(isNat(x_1)) = [[1], [1]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U13(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1], [0]] + [[1, 0], [1, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1], [0]] + [[1, 0], [1, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1], [0]] + [[1, 0], [1, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[1, 1], [0, 0]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U23(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(U106(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(U32(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U33(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[0, 1], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[0, 1], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 >>> <<< POL(U45(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U46(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U96(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[1], [0]] + [[0, 0], [0, 1]] * x_1 + [[0, 0], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 >>> <<< POL(U114(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U121(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 0]] * x_2 >>> <<< POL(U122(x_1)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[1], [1]] + [[0, 1], [0, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 + [[0, 0], [1, 0]] * x_3 + [[0, 1], [1, 0]] * x_4 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 + [[0, 0], [1, 0]] * x_3 + [[0, 1], [1, 0]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 + [[0, 0], [1, 0]] * x_3 + [[0, 1], [1, 0]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 + [[0, 0], [1, 0]] * x_3 + [[0, 1], [1, 0]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[1], [1]] + [[0, 0], [0, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 + [[0, 0], [1, 0]] * x_3 + [[0, 1], [1, 0]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 + [[0, 0], [0, 1]] * x_2 + [[0, 0], [1, 0]] * x_3 + [[0, 1], [1, 0]] * 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X cons(X1, X2) -> n__cons(X1, X2) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 0 -> n__0 take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(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)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__cons(x0, x1)) -> ISNATILISTKIND(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) 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. ---------------------------------------- (78) NO ---------------------------------------- (79) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (80) 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(activate(x0), activate(x1))), activate(y1)),U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(activate(x0), activate(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(activate(x0))), activate(y1)),U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(activate(x0))), activate(y1))) (U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)),U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1))) (U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), activate(y1)),U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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))) ---------------------------------------- (81) 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(activate(x0), activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (82) 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(activate(x0), activate(x1))),U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1)))) (U45^1(tt, n__0) -> ISNATILIST(0),U45^1(tt, n__0) -> ISNATILIST(0)) (U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))),U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0)))) (U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))),U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0)))) (U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)),U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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)) ---------------------------------------- (83) 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(activate(x0), activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(x0), activate(x1))) U45^1(tt, n__0) -> ISNATILIST(0) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (84) 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))) ---------------------------------------- (85) 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(activate(x0), activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(x0), activate(x1))) U45^1(tt, n__0) -> ISNATILIST(0) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (87) 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(activate(x0), activate(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(activate(x0), activate(x1))) U45^1(tt, n__0) -> ISNATILIST(0) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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)) ---------------------------------------- (89) 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(activate(x0), activate(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(activate(x0), activate(x1))) U45^1(tt, n__0) -> ISNATILIST(0) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (91) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) 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)) ---------------------------------------- (93) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (95) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) 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)) ---------------------------------------- (97) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (99) Obligation: Q DP problem: The TRS P consists of the following rules: U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) 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))) ---------------------------------------- (101) Obligation: Q DP problem: The TRS P consists of the following rules: U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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))) ---------------------------------------- (103) Obligation: Q DP problem: The TRS P consists of the following rules: U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (104) 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))) ---------------------------------------- (105) Obligation: Q DP problem: The TRS P consists of the following rules: U44^1(tt, n__take(x0, x1), y1) -> U45^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (106) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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))) ---------------------------------------- (109) Obligation: Q DP problem: The TRS P consists of the following rules: U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), 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(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (110) 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))) ---------------------------------------- (111) Obligation: Q DP problem: The TRS P consists of the following rules: U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) U45^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) U44^1(tt, n__length(x0), y1) -> U45^1(isNat(length(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), 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)) U45^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (112) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (113) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1))) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 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))) ---------------------------------------- (115) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1))) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (117) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1))) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(x0))), activate(y1)) U44^1(tt, n__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), activate(y1)) U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)) U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) 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(activate(x0))), activate(y1)) U44^1(tt, n__s(x0), y1) -> U45^1(isNat(s(activate(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 + 2 POL( U42^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U43^1_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 POL( U44^1_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2} POL( U45^1_2(x_1, x_2) ) = x_1 + 2x_2 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U102_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 1} POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 1} POL( U105_2(x_1, x_2) ) = 2x_2 + 1 POL( U11_2(x_1, x_2) ) = 2 POL( U111_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U112_3(x_1, ..., x_3) ) = max{0, x_1 - 1} POL( U113_3(x_1, ..., x_3) ) = max{0, x_1 - 1} POL( U114_2(x_1, x_2) ) = 1 POL( U12_2(x_1, x_2) ) = 2 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_4 - 2} POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U31_2(x_1, x_2) ) = x_2 + 1 POL( U32_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( U41_3(x_1, ..., x_3) ) = x_3 + 1 POL( U42_3(x_1, ..., x_3) ) = x_3 + 1 POL( U43_3(x_1, ..., x_3) ) = x_3 + 1 POL( U44_3(x_1, ..., x_3) ) = x_3 + 1 POL( U45_2(x_1, x_2) ) = x_2 + 1 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U91_3(x_1, ..., x_3) ) = x_3 + 1 POL( U92_3(x_1, ..., x_3) ) = x_3 + 1 POL( U93_3(x_1, ..., x_3) ) = x_3 + 1 POL( U94_3(x_1, ..., x_3) ) = x_3 + 1 POL( U95_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( length_1(x_1) ) = x_1 + 1 POL( s_1(x_1) ) = 1 POL( take_2(x_1, x_2) ) = 2x_2 POL( isNatKind_1(x_1) ) = 2x_1 + 1 POL( isNatIListKind_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( U106_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = x_1 + 1 POL( U122_1(x_1) ) = 1 POL( U13_1(x_1) ) = 2 POL( U23_1(x_1) ) = 2 POL( U33_1(x_1) ) = x_1 POL( U46_1(x_1) ) = x_1 POL( U52_1(x_1) ) = 2 POL( U62_1(x_1) ) = 2 POL( U71_1(x_1) ) = 2 POL( U81_1(x_1) ) = 2 POL( U96_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = 2x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_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 ) = 1 POL( nil ) = 1 POL( tt ) = 2 POL( U121_2(x_1, x_2) ) = max{0, x_1 - 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (119) 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(activate(x0), activate(x1))), activate(y1)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1))) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), activate(y1)) U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)) U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) 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(activate(x0), activate(x1))), activate(y1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U41^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 POL( U42^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 POL( U43^1_3(x_1, ..., x_3) ) = x_1 + 2x_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) ) = 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) ) = max{0, x_1 - 1} POL( U111_3(x_1, ..., x_3) ) = 2 POL( U112_3(x_1, ..., x_3) ) = 2 POL( U113_3(x_1, ..., x_3) ) = max{0, -2} POL( U114_2(x_1, x_2) ) = 0 POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 1} POL( U131_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_4 + 2 POL( U132_4(x_1, ..., x_4) ) = 2x_1 + 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) ) = max{0, -1} POL( U22_2(x_1, x_2) ) = 0 POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_2(x_1, x_2) ) = 0 POL( U41_3(x_1, ..., x_3) ) = max{0, -2} POL( U42_3(x_1, ..., x_3) ) = max{0, -1} POL( U43_3(x_1, ..., x_3) ) = max{0, -1} POL( U44_3(x_1, ..., x_3) ) = max{0, -1} POL( U45_2(x_1, x_2) ) = 0 POL( U51_2(x_1, x_2) ) = max{0, -1} 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) ) = 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, -1} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = 2x_2 + 2 POL( isNatKind_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatIListKind_1(x_1) ) = 0 POL( isNat_1(x_1) ) = max{0, -1} POL( U106_1(x_1) ) = 2x_1 + 2 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 2x_1 + 1 POL( U122_1(x_1) ) = 1 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) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( U96_1(x_1) ) = 0 POL( n__take_2(x_1, x_2) ) = 2x_2 + 2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 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) ) = 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNat(n__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) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (121) 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)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1))) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U45^1(tt, n__length(x0)) -> ISNATILIST(length(activate(x0))) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), activate(y1)) U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)) U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) 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(activate(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U41^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U42^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U43^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U44^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U45^1_2(x_1, x_2) ) = 2x_2 + 2 POL( ISNATILIST_1(x_1) ) = x_1 + 2 POL( U101_3(x_1, ..., x_3) ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = x_1 POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U105_2(x_1, x_2) ) = 2 POL( U11_2(x_1, x_2) ) = 2x_2 + 2 POL( U111_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U112_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U113_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U114_2(x_1, x_2) ) = 2x_2 + 1 POL( U12_2(x_1, x_2) ) = x_1 + x_2 POL( U131_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 - 2} POL( U132_4(x_1, ..., x_4) ) = 2x_2 POL( U133_4(x_1, ..., x_4) ) = 2x_2 POL( U134_4(x_1, ..., x_4) ) = 2x_2 POL( U135_4(x_1, ..., x_4) ) = 2x_2 POL( U136_4(x_1, ..., x_4) ) = 2x_2 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) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U91_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U92_3(x_1, ..., x_3) ) = 2 POL( U93_3(x_1, ..., x_3) ) = 2 POL( U94_3(x_1, ..., x_3) ) = 2 POL( U95_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = 2x_2 POL( length_1(x_1) ) = x_1 + 1 POL( s_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = x_2 POL( isNatKind_1(x_1) ) = x_1 + 1 POL( isNatIListKind_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2x_1 POL( U106_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = x_1 + 2 POL( U122_1(x_1) ) = 0 POL( U13_1(x_1) ) = x_1 POL( U23_1(x_1) ) = x_1 POL( U33_1(x_1) ) = 2 POL( U46_1(x_1) ) = 2 POL( U52_1(x_1) ) = 2 POL( U62_1(x_1) ) = 2 POL( U71_1(x_1) ) = 2 POL( U81_1(x_1) ) = x_1 POL( U96_1(x_1) ) = 2 POL( n__take_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 1 POL( 0 ) = 1 POL( n__length_1(x_1) ) = x_1 + 1 POL( n__s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 2 POL( U121_2(x_1, x_2) ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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)) 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (123) 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)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1))) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), activate(y1)) U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)) U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U45^1(tt, n__s(x0)) -> ISNATILIST(s(activate(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U41^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U42^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U43^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U44^1_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U45^1_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( ISNATILIST_1(x_1) ) = x_1 + 2 POL( U101_3(x_1, ..., x_3) ) = x_1 POL( U102_3(x_1, ..., x_3) ) = 1 POL( U103_3(x_1, ..., x_3) ) = x_1 POL( U104_3(x_1, ..., x_3) ) = 1 POL( U105_2(x_1, x_2) ) = 1 POL( U11_2(x_1, x_2) ) = 1 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) ) = 1 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) ) = 1 POL( U22_2(x_1, x_2) ) = 1 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2x_1 POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2x_1 POL( U44_3(x_1, ..., x_3) ) = 1 POL( U45_2(x_1, x_2) ) = 1 POL( U51_2(x_1, x_2) ) = 1 POL( U61_2(x_1, x_2) ) = 1 POL( U91_3(x_1, ..., x_3) ) = 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) ) = max{0, 2x_1 - 1} POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 1 POL( take_2(x_1, x_2) ) = x_2 POL( isNatKind_1(x_1) ) = 2 POL( isNatIListKind_1(x_1) ) = 1 POL( isNat_1(x_1) ) = 1 POL( U106_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = max{0, -1} POL( U13_1(x_1) ) = 1 POL( U23_1(x_1) ) = 1 POL( U33_1(x_1) ) = x_1 POL( U46_1(x_1) ) = 1 POL( U52_1(x_1) ) = max{0, 2x_1 - 1} POL( U62_1(x_1) ) = 1 POL( U71_1(x_1) ) = 1 POL( U81_1(x_1) ) = 2 POL( U96_1(x_1) ) = 1 POL( n__take_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 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 + 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 1 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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)) 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(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (125) 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)) U45^1(tt, n__take(x0, x1)) -> ISNATILIST(take(activate(x0), activate(x1))) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), activate(y1)) U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)) U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) 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(activate(x0), activate(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(U41^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(tt) = [[3A]] >>> <<< POL(U42^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(isNatKind(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(activate(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U43^1(x_1, x_2, x_3)) = [[3A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(isNatIListKind(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U44^1(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U45^1(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__take(x_1, x_2)) = [[4A]] + [[2A]] * x_1 + [[-I]] * x_2 >>> <<< POL(ISNATILIST(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(take(x_1, x_2)) = [[4A]] + [[2A]] * x_1 + [[-I]] * x_2 >>> <<< POL(n__cons(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(cons(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__zeros) = [[1A]] >>> <<< POL(0) = [[3A]] >>> <<< POL(n__0) = [[3A]] >>> <<< POL(isNat(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(zeros) = [[2A]] >>> <<< POL(n__length(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(length(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(n__s(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(n__nil) = [[3A]] >>> <<< POL(nil) = [[3A]] >>> <<< POL(U71(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U81(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U51(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U121(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U11(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U111(x_1, x_2, x_3)) = [[4A]] + [[2A]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(isNatList(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U52(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U62(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U92(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U93(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U94(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U95(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U13(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[4A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[4A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[4A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[4A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U106(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U32(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U33(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[1A]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U45(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U46(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U96(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U114(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[2A]] * x_2 >>> <<< POL(U122(x_1)) = [[4A]] + [[-I]] * x_1 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[5A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[1A]] + [[2A]] * x_1 + [[-I]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[3A]] * x_3 + [[-I]] * x_4 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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__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) cons(X1, X2) -> n__cons(X1, X2) 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)) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(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(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (127) 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)) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U45^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(activate(x0), x1)) U45^1(tt, x0) -> ISNATILIST(x0) 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__cons(x0, x1), y1) -> U45^1(isNat(cons(activate(x0), x1)), activate(y1)) U44^1(tt, x0, y1) -> U45^1(isNat(x0), activate(y1)) U44^1(tt, n__0, y0) -> U45^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) U105(tt, V2) -> U106(isNatIList(activate(V2))) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) U114(tt, L) -> s(length(activate(L))) U12(tt, V1) -> U13(isNatList(activate(V1))) U121(tt, IL) -> U122(isNatIListKind(activate(IL))) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) U45(tt, V2) -> U46(isNatIList(activate(V2))) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) U95(tt, V2) -> U96(isNatList(activate(V2))) U96(tt) -> tt isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (128) 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, V1, V2) -> U95^1(isNat(activate(V1)), activate(V2)) U95^1(tt, V2) -> ISNATLIST(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U94^1(tt, 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(activate(x0), activate(x1))), activate(y1)),U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(activate(x0), activate(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(activate(x0))), activate(y1)),U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1))) (U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)),U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1))) (U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), activate(y1)),U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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))) ---------------------------------------- (130) 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, V2) -> ISNATLIST(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(activate(x0), activate(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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) 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(activate(x0), activate(x1))),U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1)))) (U95^1(tt, n__0) -> ISNATLIST(0),U95^1(tt, n__0) -> ISNATLIST(0)) (U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))),U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0)))) (U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))),U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0)))) (U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)),U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(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)) ---------------------------------------- (132) 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__zeros, y1) -> U95^1(isNat(zeros), activate(y1)) U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(activate(x0), activate(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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(x0), activate(x1))) U95^1(tt, n__0) -> ISNATLIST(0) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U94^1(tt, n__zeros, 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))) ---------------------------------------- (134) 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(activate(x0), activate(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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(x0), activate(x1))) U95^1(tt, n__0) -> ISNATLIST(0) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__zeros) -> ISNATLIST(zeros) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) U95^1(tt, n__0) -> ISNATLIST(0) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, n__nil) -> ISNATLIST(nil) U95^1(tt, x0) -> ISNATLIST(x0) U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1)) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 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)) ---------------------------------------- (138) 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(activate(x0), activate(x1))), activate(y1)) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) U95^1(tt, n__0) -> ISNATLIST(0) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, n__nil) -> ISNATLIST(nil) U95^1(tt, x0) -> ISNATLIST(x0) U94^1(tt, n__0, y1) -> U95^1(isNat(0), activate(y1)) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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__0) -> ISNATLIST(0) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) 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)) ---------------------------------------- (142) 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (144) 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) 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)) ---------------------------------------- (146) 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) 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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 1 SCC with 1 less node. ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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__0, y1) -> U95^1(isNat(0), activate(y1)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) 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))) ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) U95^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) 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))) ---------------------------------------- (152) Obligation: Q DP problem: The TRS P consists of the following rules: U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(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)) U94^1(tt, n__0, y0) -> U95^1(isNat(n__0), 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (153) 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))) ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: U94^1(tt, n__take(x0, x1), y1) -> U95^1(isNat(take(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (155) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (156) Obligation: Q DP problem: The TRS P consists of the following rules: U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (157) 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))) ---------------------------------------- (158) Obligation: Q DP problem: The TRS P consists of the following rules: U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (159) 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))) ---------------------------------------- (160) Obligation: Q DP problem: The TRS P consists of the following rules: U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) U95^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) U94^1(tt, n__length(x0), y1) -> U95^1(isNat(length(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) U95^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (161) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (162) 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (163) 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))) ---------------------------------------- (164) 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (165) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (166) 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(x0))), activate(y1)) U94^1(tt, n__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (167) 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(activate(x0))), activate(y1)) U94^1(tt, n__s(x0), y1) -> U95^1(isNat(s(activate(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 + 1 POL( U92^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 POL( U93^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 POL( U94^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 POL( U95^1_2(x_1, x_2) ) = 2x_2 + 1 POL( ISNATLIST_1(x_1) ) = 2x_1 + 1 POL( U101_3(x_1, ..., x_3) ) = 1 POL( U102_3(x_1, ..., x_3) ) = 1 POL( U103_3(x_1, ..., x_3) ) = 1 POL( U104_3(x_1, ..., x_3) ) = 1 POL( U105_2(x_1, x_2) ) = 1 POL( U11_2(x_1, x_2) ) = 1 POL( U111_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 1} POL( U112_3(x_1, ..., x_3) ) = x_1 + 2x_3 POL( U113_3(x_1, ..., x_3) ) = x_1 POL( U114_2(x_1, x_2) ) = 1 POL( U12_2(x_1, x_2) ) = 1 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) ) = 1 POL( U22_2(x_1, x_2) ) = 1 POL( U31_2(x_1, x_2) ) = x_1 + 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = x_1 + x_3 + 1 POL( U42_3(x_1, ..., x_3) ) = x_3 + 1 POL( U43_3(x_1, ..., x_3) ) = x_1 + 1 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( U61_2(x_1, x_2) ) = 2x_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) ) = x_1 + 1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( length_1(x_1) ) = x_1 + 1 POL( s_1(x_1) ) = 1 POL( take_2(x_1, x_2) ) = 2x_2 POL( isNatKind_1(x_1) ) = x_1 + 1 POL( isNatIListKind_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 1 POL( U106_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = 2 POL( U122_1(x_1) ) = max{0, 2x_1 - 1} POL( U13_1(x_1) ) = 1 POL( U23_1(x_1) ) = 1 POL( U33_1(x_1) ) = 1 POL( U46_1(x_1) ) = 2 POL( U52_1(x_1) ) = x_1 POL( U62_1(x_1) ) = x_1 POL( U71_1(x_1) ) = 1 POL( U81_1(x_1) ) = 1 POL( U96_1(x_1) ) = 1 POL( n__take_2(x_1, x_2) ) = 2x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_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 ) = 1 POL( nil ) = 1 POL( tt ) = 1 POL( U121_2(x_1, x_2) ) = 2x_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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 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(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (168) 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(activate(x0), activate(x1))), activate(y1)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (169) 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(activate(x0), activate(x1))), activate(y1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U91^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U92^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U93^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U94^1_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( U95^1_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( ISNATLIST_1(x_1) ) = 2x_1 POL( U101_3(x_1, ..., x_3) ) = x_1 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2 POL( U105_2(x_1, x_2) ) = 2 POL( U11_2(x_1, x_2) ) = 2x_2 + 2 POL( U111_3(x_1, ..., x_3) ) = x_1 POL( U112_3(x_1, ..., x_3) ) = 2 POL( U113_3(x_1, ..., x_3) ) = 2 POL( U114_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U12_2(x_1, x_2) ) = 2x_2 + 2 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) ) = max{0, 2x_1 - 2} POL( U22_2(x_1, x_2) ) = 2 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2} POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2} POL( U93_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2} POL( U94_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2} POL( U95_2(x_1, x_2) ) = x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( length_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 2 POL( take_2(x_1, x_2) ) = 2x_2 + 2 POL( isNatKind_1(x_1) ) = 2 POL( isNatIListKind_1(x_1) ) = 2 POL( isNat_1(x_1) ) = x_1 + 2 POL( U106_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = x_1 POL( U122_1(x_1) ) = 2 POL( U13_1(x_1) ) = 2x_1 POL( U23_1(x_1) ) = 2 POL( U33_1(x_1) ) = 2 POL( U46_1(x_1) ) = x_1 POL( U52_1(x_1) ) = 2 POL( U62_1(x_1) ) = 2 POL( U71_1(x_1) ) = 2 POL( U81_1(x_1) ) = 2 POL( U96_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = 2x_2 + 2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 POL( n__s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 2 POL( nil ) = 2 POL( tt ) = 2 POL( U121_2(x_1, x_2) ) = 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (170) 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)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__length(x0)) -> ISNATLIST(length(activate(x0))) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (171) 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(activate(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U91^1_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U92^1_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U93^1_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U94^1_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U95^1_2(x_1, x_2) ) = 2x_2 + 1 POL( ISNATLIST_1(x_1) ) = x_1 + 1 POL( U101_3(x_1, ..., x_3) ) = 1 POL( U102_3(x_1, ..., x_3) ) = 1 POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U104_3(x_1, ..., x_3) ) = 1 POL( U105_2(x_1, x_2) ) = 1 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) ) = 0 POL( U114_2(x_1, x_2) ) = max{0, -2} POL( U12_2(x_1, x_2) ) = 1 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_4 - 1} POL( U133_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 1 POL( U31_2(x_1, x_2) ) = 1 POL( U32_2(x_1, x_2) ) = 1 POL( U41_3(x_1, ..., x_3) ) = x_2 + 2 POL( U42_3(x_1, ..., x_3) ) = x_2 + 2 POL( U43_3(x_1, ..., x_3) ) = x_2 + 2 POL( U44_3(x_1, ..., x_3) ) = x_1 + x_2 + 1 POL( U45_2(x_1, x_2) ) = x_1 POL( U51_2(x_1, x_2) ) = 1 POL( U61_2(x_1, x_2) ) = 1 POL( U91_3(x_1, ..., x_3) ) = 1 POL( U92_3(x_1, ..., x_3) ) = 1 POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U94_3(x_1, ..., x_3) ) = 1 POL( U95_2(x_1, x_2) ) = 1 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( length_1(x_1) ) = x_1 + 2 POL( s_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = x_2 POL( isNatKind_1(x_1) ) = 2x_1 + 2 POL( isNatIListKind_1(x_1) ) = 1 POL( isNat_1(x_1) ) = x_1 + 2 POL( U106_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 1 POL( U122_1(x_1) ) = 0 POL( U13_1(x_1) ) = x_1 POL( U23_1(x_1) ) = 1 POL( U33_1(x_1) ) = 1 POL( U46_1(x_1) ) = 1 POL( U52_1(x_1) ) = 1 POL( U62_1(x_1) ) = 1 POL( U71_1(x_1) ) = 1 POL( U81_1(x_1) ) = 1 POL( U96_1(x_1) ) = 1 POL( n__take_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = x_1 + 2 POL( n__s_1(x_1) ) = 0 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 1 POL( U121_2(x_1, x_2) ) = max{0, -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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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)) 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(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (172) 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)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(x0))) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (173) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U95^1(tt, n__s(x0)) -> ISNATLIST(s(activate(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) ) = x_2 + 2x_3 + 1 POL( U92^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 1 POL( U93^1_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 1 POL( U94^1_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 POL( U95^1_2(x_1, x_2) ) = x_1 + 2x_2 POL( ISNATLIST_1(x_1) ) = x_1 + 1 POL( U101_3(x_1, ..., x_3) ) = 1 POL( U102_3(x_1, ..., x_3) ) = 1 POL( U103_3(x_1, ..., x_3) ) = x_1 POL( U104_3(x_1, ..., x_3) ) = x_1 POL( U105_2(x_1, x_2) ) = 1 POL( U11_2(x_1, x_2) ) = 1 POL( U111_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 1 POL( U112_3(x_1, ..., x_3) ) = x_1 + 1 POL( U113_3(x_1, ..., x_3) ) = 2 POL( U114_2(x_1, x_2) ) = 2 POL( U12_2(x_1, x_2) ) = 1 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + x_4 - 2} POL( U133_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U134_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U135_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = 1 POL( U31_2(x_1, x_2) ) = 1 POL( U32_2(x_1, x_2) ) = 1 POL( U41_3(x_1, ..., x_3) ) = 1 POL( U42_3(x_1, ..., x_3) ) = 1 POL( U43_3(x_1, ..., x_3) ) = x_1 POL( U44_3(x_1, ..., x_3) ) = x_1 POL( U45_2(x_1, x_2) ) = 1 POL( U51_2(x_1, x_2) ) = 1 POL( U61_2(x_1, x_2) ) = 1 POL( U91_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 POL( U92_3(x_1, ..., x_3) ) = x_1 + 2x_3 POL( U93_3(x_1, ..., x_3) ) = 2x_3 + 1 POL( U94_3(x_1, ..., x_3) ) = x_1 + 2x_3 POL( U95_2(x_1, x_2) ) = 2x_2 + 1 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( length_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = 1 POL( take_2(x_1, x_2) ) = x_2 POL( isNatKind_1(x_1) ) = x_1 + 1 POL( isNatIListKind_1(x_1) ) = 1 POL( isNat_1(x_1) ) = 1 POL( U106_1(x_1) ) = max{0, 2x_1 - 1} POL( isNatIList_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = 2x_1 + 1 POL( U122_1(x_1) ) = max{0, -2} POL( U13_1(x_1) ) = 1 POL( U23_1(x_1) ) = 1 POL( U33_1(x_1) ) = 1 POL( U46_1(x_1) ) = x_1 POL( U52_1(x_1) ) = x_1 POL( U62_1(x_1) ) = 1 POL( U71_1(x_1) ) = x_1 POL( U81_1(x_1) ) = 2 POL( U96_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( n__s_1(x_1) ) = 1 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( tt ) = 1 POL( U121_2(x_1, x_2) ) = max{0, x_1 - 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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)) 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (174) 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)) U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (175) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U95^1(tt, n__take(x0, x1)) -> ISNATLIST(take(activate(x0), activate(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(U91^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(tt) = [[3A]] >>> <<< POL(U92^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(isNatKind(x_1)) = [[1A]] + [[1A]] * x_1 >>> <<< POL(activate(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U93^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(isNatIListKind(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(U94^1(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U95^1(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__take(x_1, x_2)) = [[3A]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(ISNATLIST(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(take(x_1, x_2)) = [[3A]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__cons(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(cons(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__zeros) = [[0A]] >>> <<< POL(0) = [[2A]] >>> <<< POL(n__0) = [[2A]] >>> <<< POL(isNat(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(zeros) = [[2A]] >>> <<< POL(n__length(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(length(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(n__s(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(n__nil) = [[3A]] >>> <<< POL(nil) = [[3A]] >>> <<< POL(U71(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(U81(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U51(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U121(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[-I]] + [[1A]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U11(x_1, x_2)) = [[4A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U111(x_1, x_2, x_3)) = [[2A]] + [[2A]] * x_1 + [[2A]] * x_2 + [[0A]] * x_3 >>> <<< POL(isNatList(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U52(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U62(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U92(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U93(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U94(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U95(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[4A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U13(x_1)) = [[4A]] + [[-I]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U23(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U106(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U32(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U33(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U45(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U46(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U96(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[3A]] + [[2A]] * x_1 + [[2A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[5A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U114(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[2A]] * x_2 >>> <<< POL(U122(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[4A]] + [[1A]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[4A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[3A]] + [[1A]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[4A]] + [[0A]] * x_1 + [[2A]] * x_2 + [[2A]] * x_3 + [[1A]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[4A]] + [[0A]] * x_1 + [[2A]] * 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: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) cons(X1, X2) -> n__cons(X1, X2) 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)) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (176) 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)) ISNATLIST(n__cons(V1, V2)) -> U91^1(isNatKind(activate(V1)), activate(V1), activate(V2)) U95^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(activate(x0), x1)) U95^1(tt, x0) -> ISNATLIST(x0) 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__cons(x0, x1), y1) -> U95^1(isNat(cons(activate(x0), x1)), 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)) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (177) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (178) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U111^1(tt, L, N) -> U112^1(isNatIListKind(activate(L)), 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) ) = x_1 POL( U111^1_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 1} POL( U112^1_3(x_1, ..., x_3) ) = x_2 POL( U113^1_3(x_1, ..., x_3) ) = x_2 POL( U114^1_2(x_1, x_2) ) = x_2 POL( U101_3(x_1, ..., x_3) ) = x_2 + x_3 + 1 POL( U102_3(x_1, ..., x_3) ) = x_2 + x_3 + 1 POL( U103_3(x_1, ..., x_3) ) = x_2 + x_3 + 1 POL( U104_3(x_1, ..., x_3) ) = x_2 + x_3 + 1 POL( U105_2(x_1, x_2) ) = x_1 + x_2 POL( U11_2(x_1, x_2) ) = x_2 + 1 POL( U111_3(x_1, ..., x_3) ) = 2x_2 POL( U112_3(x_1, ..., x_3) ) = 2x_2 POL( U113_3(x_1, ..., x_3) ) = 2x_2 POL( U114_2(x_1, x_2) ) = 2x_2 POL( U12_2(x_1, x_2) ) = x_2 + 1 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) ) = x_2 + 1 POL( U22_2(x_1, x_2) ) = x_2 + 1 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U42_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U43_3(x_1, ..., x_3) ) = x_1 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = x_1 POL( U91_3(x_1, ..., x_3) ) = 2x_3 + 1 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) ) = x_1 + 1 POL( length_1(x_1) ) = x_1 POL( s_1(x_1) ) = 2x_1 POL( take_2(x_1, x_2) ) = x_1 + x_2 POL( isNatIListKind_1(x_1) ) = x_1 + 2 POL( isNat_1(x_1) ) = x_1 + 1 POL( isNatKind_1(x_1) ) = x_1 + 2 POL( U106_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( U122_1(x_1) ) = 1 POL( U23_1(x_1) ) = x_1 POL( U46_1(x_1) ) = 2 POL( U52_1(x_1) ) = 2 POL( U62_1(x_1) ) = 2 POL( U71_1(x_1) ) = 2 POL( U81_1(x_1) ) = 2 POL( n__take_2(x_1, x_2) ) = x_1 + x_2 POL( U13_1(x_1) ) = x_1 POL( U33_1(x_1) ) = 2 POL( U96_1(x_1) ) = max{0, 2x_1 - 2} POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 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 ) = 2 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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__nil) -> tt isNatIListKind(n__zeros) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (179) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U111^1(isNatList(activate(L)), activate(L), 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (180) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. ---------------------------------------- (181) TRUE ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: 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) ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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)) ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1)) ISNATKIND(n__s(V1)) -> ACTIVATE(V1) U135^1(tt, IL, M, N) -> ACTIVATE(N) U135^1(tt, IL, M, N) -> ACTIVATE(IL) U135^1(tt, IL, M, N) -> ACTIVATE(M) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (183) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U134^1(tt, IL, M, N) -> U135^1(isNat(activate(N)), activate(IL), activate(M), activate(N)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNATKIND_1(x_1) ) = x_1 + 1 POL( TAKE_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U131^1_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + 2x_4 POL( U132^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( U133^1_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 + 2x_4 - 2} POL( U134^1_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4 + 2 POL( U135^1_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + x_4 - 1} POL( U136^1_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + x_3 + x_4 - 1} POL( U101_3(x_1, ..., x_3) ) = x_1 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U105_2(x_1, x_2) ) = 2 POL( U11_2(x_1, x_2) ) = 2 POL( U111_3(x_1, ..., x_3) ) = x_2 + 1 POL( U112_3(x_1, ..., x_3) ) = x_2 + 1 POL( U113_3(x_1, ..., x_3) ) = x_2 + 1 POL( U114_2(x_1, x_2) ) = x_2 + 1 POL( U12_2(x_1, x_2) ) = 2 POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1 POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1 POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1 POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + 2x_4 - 1} POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1 POL( U136_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + 2x_4 - 1} POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 2 POL( U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( U93_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( U94_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( U95_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = 2 POL( length_1(x_1) ) = x_1 + 1 POL( s_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( isNatIListKind_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( isNatKind_1(x_1) ) = 2 POL( U106_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2x_1 POL( U122_1(x_1) ) = max{0, x_1 - 1} POL( U13_1(x_1) ) = 2 POL( U23_1(x_1) ) = 2 POL( U33_1(x_1) ) = 2 POL( U52_1(x_1) ) = 2 POL( U62_1(x_1) ) = 2 POL( U71_1(x_1) ) = 2 POL( U81_1(x_1) ) = 2 POL( U96_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( U46_1(x_1) ) = x_1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = x_1 + 1 POL( n__s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 1 POL( nil ) = 1 POL( U121_2(x_1, x_2) ) = 1 POL( tt ) = 2 POL( ACTIVATE_1(x_1) ) = x_1 + 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X 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)) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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 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 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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: 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)) 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) ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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)) ISNATKIND(n__s(V1)) -> ISNATKIND(activate(V1)) ISNATKIND(n__s(V1)) -> ACTIVATE(V1) U135^1(tt, IL, M, N) -> ACTIVATE(N) U135^1(tt, IL, M, N) -> ACTIVATE(IL) U135^1(tt, IL, M, N) -> ACTIVATE(M) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (185) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 14 less nodes. ---------------------------------------- (186) Complex Obligation (AND) ---------------------------------------- (187) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__s(X)) -> ACTIVATE(X) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (188) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (189) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__s(X)) -> ACTIVATE(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (190) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) The graph contains the following edges 1 > 1 *ACTIVATE(n__s(X)) -> ACTIVATE(X) The graph contains the following edges 1 > 1 ---------------------------------------- (191) YES ---------------------------------------- (192) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (193) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 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)) = [[0A]] + [[3A]] * 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)) = [[4A]] + [[3A]] * x_1 + [[-I]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[4A]] + [[3A]] * x_1 + [[-I]] * x_2 >>> <<< POL(n__0) = [[2A]] >>> <<< POL(0) = [[2A]] >>> <<< POL(n__length(x_1)) = [[4A]] + [[2A]] * x_1 >>> <<< POL(length(x_1)) = [[4A]] + [[2A]] * 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)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__nil) = [[3A]] >>> <<< POL(nil) = [[3A]] >>> <<< POL(U111(x_1, x_2, x_3)) = [[3A]] + [[3A]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 >>> <<< POL(isNatList(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(tt) = [[3A]] >>> <<< POL(U91(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(isNatKind(x_1)) = [[0A]] + [[1A]] * x_1 >>> <<< POL(U71(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(isNatIListKind(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U51(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U81(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(U52(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U62(x_1)) = [[3A]] + [[-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)) = [[0A]] + [[1A]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[3A]] + [[2A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U12(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U13(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[4A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[4A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[4A]] + [[0A]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[2A]] + [[1A]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U105(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U21(x_1, x_2)) = [[3A]] + [[1A]] * x_1 + [[2A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[4A]] + [[1A]] * x_1 + [[2A]] * x_2 >>> <<< POL(U23(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U106(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U32(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U33(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U41(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U42(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U43(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U44(x_1, x_2, x_3)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U45(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U46(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U96(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U112(x_1, x_2, x_3)) = [[3A]] + [[3A]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U113(x_1, x_2, x_3)) = [[5A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U114(x_1, x_2)) = [[5A]] + [[-I]] * x_1 + [[3A]] * x_2 >>> <<< POL(U121(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U122(x_1)) = [[3A]] + [[-I]] * x_1 >>> <<< POL(U131(x_1, x_2, x_3, x_4)) = [[5A]] + [[2A]] * x_1 + [[-I]] * x_2 + [[4A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U132(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[4A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U133(x_1, x_2, x_3, x_4)) = [[4A]] + [[2A]] * x_1 + [[-I]] * x_2 + [[4A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U134(x_1, x_2, x_3, x_4)) = [[5A]] + [[2A]] * x_1 + [[-I]] * x_2 + [[4A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U135(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[4A]] * x_3 + [[-I]] * x_4 >>> <<< POL(U136(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[4A]] * x_3 + [[-I]] * x_4 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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))) isNatIListKind(n__nil) -> tt isNatIListKind(n__zeros) -> tt U71(tt) -> tt isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) U81(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(activate(V2))) U52(tt) -> tt isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) U61(tt, V2) -> U62(isNatIListKind(activate(V2))) U62(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) U12(tt, V1) -> U13(isNatList(activate(V1))) U13(tt) -> tt isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U105(tt, V2) -> U106(isNatIList(activate(V2))) isNatIList(n__zeros) -> tt U106(tt) -> tt isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) U32(tt, V) -> U33(isNatList(activate(V))) U33(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))) take(0, IL) -> U121(isNatIList(IL), IL) 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))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (194) 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (195) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (196) YES