/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- NO proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 122 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 40 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 23 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 25 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 0 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) UsableRulesReductionPairsProof [EQUIVALENT, 30 ms] (15) QDP (16) MRRProof [EQUIVALENT, 23 ms] (17) QDP (18) DependencyGraphProof [EQUIVALENT, 0 ms] (19) AND (20) QDP (21) TransformationProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) DependencyGraphProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) DependencyGraphProof [EQUIVALENT, 0 ms] (32) QDP (33) TransformationProof [EQUIVALENT, 0 ms] (34) QDP (35) 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) TransformationProof [EQUIVALENT, 0 ms] (48) QDP (49) DependencyGraphProof [EQUIVALENT, 0 ms] (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) DependencyGraphProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) TransformationProof [EQUIVALENT, 0 ms] (62) QDP (63) DependencyGraphProof [EQUIVALENT, 0 ms] (64) QDP (65) TransformationProof [EQUIVALENT, 0 ms] (66) QDP (67) DependencyGraphProof [EQUIVALENT, 0 ms] (68) QDP (69) TransformationProof [EQUIVALENT, 0 ms] (70) QDP (71) MRRProof [EQUIVALENT, 36 ms] (72) QDP (73) MRRProof [EQUIVALENT, 0 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 57 ms] (76) QDP (77) NonTerminationLoopProof [COMPLETE, 0 ms] (78) NO (79) QDP (80) QDPOrderProof [EQUIVALENT, 63 ms] (81) QDP (82) DependencyGraphProof [EQUIVALENT, 0 ms] (83) TRUE (84) QDP (85) QDPOrderProof [EQUIVALENT, 47 ms] (86) QDP (87) PisEmptyProof [EQUIVALENT, 0 ms] (88) YES (89) QDP (90) UsableRulesReductionPairsProof [EQUIVALENT, 25 ms] (91) QDP (92) TransformationProof [EQUIVALENT, 0 ms] (93) QDP (94) TransformationProof [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) DependencyGraphProof [EQUIVALENT, 0 ms] (103) QDP (104) TransformationProof [EQUIVALENT, 0 ms] (105) QDP (106) TransformationProof [EQUIVALENT, 0 ms] (107) QDP (108) DependencyGraphProof [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) TransformationProof [EQUIVALENT, 0 ms] (117) QDP (118) TransformationProof [EQUIVALENT, 0 ms] (119) QDP (120) DependencyGraphProof [EQUIVALENT, 0 ms] (121) QDP (122) TransformationProof [EQUIVALENT, 0 ms] (123) QDP (124) DependencyGraphProof [EQUIVALENT, 0 ms] (125) QDP (126) TransformationProof [EQUIVALENT, 0 ms] (127) QDP (128) DependencyGraphProof [EQUIVALENT, 0 ms] (129) QDP (130) TransformationProof [EQUIVALENT, 0 ms] (131) QDP (132) TransformationProof [EQUIVALENT, 0 ms] (133) QDP (134) DependencyGraphProof [EQUIVALENT, 0 ms] (135) QDP (136) TransformationProof [EQUIVALENT, 0 ms] (137) QDP (138) DependencyGraphProof [EQUIVALENT, 0 ms] (139) QDP (140) TransformationProof [EQUIVALENT, 0 ms] (141) QDP (142) MRRProof [EQUIVALENT, 11 ms] (143) QDP (144) QDPOrderProof [EQUIVALENT, 31 ms] (145) QDP (146) NonTerminationLoopProof [COMPLETE, 0 ms] (147) NO ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt) -> tt U21(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = x_1 + 2*x_2 POL(U42(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = x_1 POL(n__nil) = 2 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 2 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isNatList(n__nil) -> tt length(nil) -> 0 ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt) -> tt U21(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1)) = 1 + 2*x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 2*x_1 POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 POL(U62(x_1, x_2)) = 1 + x_1 + x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = 2*x_1 POL(isNatIList(x_1)) = 2*x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 1 + x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 1 + x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U11(tt) -> tt isNat(n__length(V1)) -> U11(isNatList(activate(V1))) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U21(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 1 + 2*x_1 POL(U41(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = 1 + 2*x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 2*x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U31(tt) -> tt isNatIList(n__zeros) -> tt ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U21(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = 2*x_1 POL(isNatIList(x_1)) = 1 + x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 2*x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isNatIList(V) -> U31(isNatList(activate(V))) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U21(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: ZEROS -> CONS(0, n__zeros) ZEROS -> 0^1 U41^1(tt, V2) -> U42^1(isNatIList(activate(V2))) U41^1(tt, V2) -> ISNATILIST(activate(V2)) U41^1(tt, V2) -> ACTIVATE(V2) U51^1(tt, V2) -> U52^1(isNatList(activate(V2))) U51^1(tt, V2) -> ISNATLIST(activate(V2)) U51^1(tt, V2) -> ACTIVATE(V2) U61^1(tt, L, N) -> U62^1(isNat(activate(N)), activate(L)) U61^1(tt, L, N) -> ISNAT(activate(N)) U61^1(tt, L, N) -> ACTIVATE(N) U61^1(tt, L, N) -> ACTIVATE(L) U62^1(tt, L) -> S(length(activate(L))) U62^1(tt, L) -> LENGTH(activate(L)) U62^1(tt, L) -> ACTIVATE(L) ISNAT(n__s(V1)) -> U21^1(isNat(activate(V1))) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNat(activate(V1)), activate(V2)) ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATLIST(n__cons(V1, V2)) -> U51^1(isNat(activate(V1)), activate(V2)) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) LENGTH(cons(N, L)) -> U61^1(isNatList(activate(L)), activate(L), N) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) LENGTH(cons(N, L)) -> ACTIVATE(L) ACTIVATE(n__zeros) -> ZEROS ACTIVATE(n__0) -> 0^1 ACTIVATE(n__length(X)) -> LENGTH(X) ACTIVATE(n__s(X)) -> S(X) ACTIVATE(n__cons(X1, X2)) -> CONS(X1, X2) ACTIVATE(n__nil) -> NIL The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U21(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 15 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__length(X)) -> LENGTH(X) LENGTH(cons(N, L)) -> U61^1(isNatList(activate(L)), activate(L), N) U61^1(tt, L, N) -> U62^1(isNat(activate(N)), activate(L)) U62^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) ISNATLIST(n__cons(V1, V2)) -> U51^1(isNat(activate(V1)), activate(V2)) U51^1(tt, V2) -> ISNATLIST(activate(V2)) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) U51^1(tt, V2) -> ACTIVATE(V2) LENGTH(cons(N, L)) -> ACTIVATE(L) U62^1(tt, L) -> ACTIVATE(L) U61^1(tt, L, N) -> ISNAT(activate(N)) U61^1(tt, L, N) -> ACTIVATE(N) U61^1(tt, L, N) -> ACTIVATE(L) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U21(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = x_1 POL(ISNAT(x_1)) = x_1 POL(ISNATLIST(x_1)) = 2*x_1 POL(LENGTH(x_1)) = 2*x_1 POL(U21(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U51^1(x_1, x_2)) = x_1 + 2*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U61^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + 2*x_2 POL(U62^1(x_1, x_2)) = 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 2*x_1 POL(n__nil) = 2 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 2 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__length(X)) -> LENGTH(X) LENGTH(cons(N, L)) -> U61^1(isNatList(activate(L)), activate(L), N) U61^1(tt, L, N) -> U62^1(isNat(activate(N)), activate(L)) U62^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) ISNATLIST(n__cons(V1, V2)) -> U51^1(isNat(activate(V1)), activate(V2)) U51^1(tt, V2) -> ISNATLIST(activate(V2)) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) U51^1(tt, V2) -> ACTIVATE(V2) LENGTH(cons(N, L)) -> ACTIVATE(L) U62^1(tt, L) -> ACTIVATE(L) U61^1(tt, L, N) -> ISNAT(activate(N)) U61^1(tt, L, N) -> ACTIVATE(N) U61^1(tt, L, N) -> ACTIVATE(L) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) U51^1(tt, V2) -> ACTIVATE(V2) LENGTH(cons(N, L)) -> ACTIVATE(L) U62^1(tt, L) -> ACTIVATE(L) U61^1(tt, L, N) -> ISNAT(activate(N)) U61^1(tt, L, N) -> ACTIVATE(N) U61^1(tt, L, N) -> ACTIVATE(L) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = x_1 POL(ISNAT(x_1)) = x_1 POL(ISNATLIST(x_1)) = 1 + x_1 POL(LENGTH(x_1)) = 2 + x_1 POL(U21(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U51^1(x_1, x_2)) = 1 + x_1 + x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 POL(U61^1(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 POL(U62(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(U62^1(x_1, x_2)) = 2 + x_1 + x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2 + x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 2 + x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__length(X)) -> LENGTH(X) LENGTH(cons(N, L)) -> U61^1(isNatList(activate(L)), activate(L), N) U61^1(tt, L, N) -> U62^1(isNat(activate(N)), activate(L)) U62^1(tt, L) -> LENGTH(activate(L)) ISNATLIST(n__cons(V1, V2)) -> U51^1(isNat(activate(V1)), activate(V2)) U51^1(tt, V2) -> ISNATLIST(activate(V2)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes. ---------------------------------------- (19) Complex Obligation (AND) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, V2) -> ISNATLIST(activate(V2)) ISNATLIST(n__cons(V1, V2)) -> U51^1(isNat(activate(V1)), activate(V2)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U51^1(tt, V2) -> ISNATLIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__zeros) -> ISNATLIST(zeros),U51^1(tt, n__zeros) -> ISNATLIST(zeros)) (U51^1(tt, n__0) -> ISNATLIST(0),U51^1(tt, n__0) -> ISNATLIST(0)) (U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)),U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0))) (U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)),U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0))) (U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)),U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1))) (U51^1(tt, n__nil) -> ISNATLIST(nil),U51^1(tt, n__nil) -> ISNATLIST(nil)) (U51^1(tt, x0) -> ISNATLIST(x0),U51^1(tt, x0) -> ISNATLIST(x0)) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(V1, V2)) -> U51^1(isNat(activate(V1)), activate(V2)) U51^1(tt, n__zeros) -> ISNATLIST(zeros) U51^1(tt, n__0) -> ISNATLIST(0) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) U51^1(tt, n__nil) -> ISNATLIST(nil) U51^1(tt, x0) -> ISNATLIST(x0) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(V1, V2)) -> U51^1(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__zeros, y1)) -> U51^1(isNat(zeros), activate(y1)),ISNATLIST(n__cons(n__zeros, y1)) -> U51^1(isNat(zeros), activate(y1))) (ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1)),ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1))) (ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)),ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1))) (ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)),ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1))) (ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)),ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1))) (ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)),ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1))) (ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)),ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__zeros) -> ISNATLIST(zeros) U51^1(tt, n__0) -> ISNATLIST(0) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) U51^1(tt, n__nil) -> ISNATLIST(nil) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(n__zeros, y1)) -> U51^1(isNat(zeros), activate(y1)) ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U51^1(tt, n__zeros) -> ISNATLIST(zeros) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)),U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros))) (U51^1(tt, n__zeros) -> ISNATLIST(n__zeros),U51^1(tt, n__zeros) -> ISNATLIST(n__zeros)) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__0) -> ISNATLIST(0) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) U51^1(tt, n__nil) -> ISNATLIST(nil) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(n__zeros, y1)) -> U51^1(isNat(zeros), activate(y1)) ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) U51^1(tt, n__zeros) -> ISNATLIST(n__zeros) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> U51^1(isNat(zeros), activate(y1)) U51^1(tt, n__0) -> ISNATLIST(0) ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y1)) -> U51^1(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)),ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0))) (ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__zeros), activate(y0)),ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__zeros), activate(y0))) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__0) -> ISNATLIST(0) ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__zeros), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1)) U51^1(tt, n__0) -> ISNATLIST(0) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__0, y1)) -> U51^1(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)),ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__0) -> ISNATLIST(0) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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__0) -> ISNATLIST(0) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__0) -> ISNATLIST(n__0),U51^1(tt, n__0) -> ISNATLIST(n__0)) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__0) -> ISNATLIST(n__0) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__s(x0)) -> ISNATLIST(s(x0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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__s(x0)) -> ISNATLIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__s(x0)) -> ISNATLIST(n__s(x0)),U51^1(tt, n__s(x0)) -> ISNATLIST(n__s(x0))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__s(x0)) -> ISNATLIST(n__s(x0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)),ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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__cons(x0, x1)) -> ISNATLIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)),U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> U51^1(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__cons(x0, x1), y2)) -> U51^1(isNat(n__cons(x0, x1)), activate(y2)),ISNATLIST(n__cons(n__cons(x0, x1), y2)) -> U51^1(isNat(n__cons(x0, x1)), activate(y2))) ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__cons(x0, x1), y2)) -> U51^1(isNat(n__cons(x0, x1)), activate(y2)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(nil) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U51^1(tt, n__nil) -> ISNATLIST(nil) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__nil) -> ISNATLIST(n__nil),U51^1(tt, n__nil) -> ISNATLIST(n__nil)) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__nil) -> ISNATLIST(n__nil) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__nil, y1)) -> U51^1(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__nil, y0)) -> U51^1(isNat(n__nil), activate(y0)),ISNATLIST(n__cons(n__nil, y0)) -> U51^1(isNat(n__nil), activate(y0))) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) ISNATLIST(n__cons(n__nil, y0)) -> U51^1(isNat(n__nil), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U51^1(tt, n__zeros) -> ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)),U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros))) (U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)),U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros))) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__cons(0, n__zeros)), activate(y0)),ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__cons(0, n__zeros)), activate(y0))) (ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(n__0, n__zeros)), activate(y0)),ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(n__0, n__zeros)), activate(y0))) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__cons(0, n__zeros)), activate(y0)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(n__0, n__zeros)), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(n__0, n__zeros)), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__cons(n__0, n__zeros)), activate(y0)),ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__cons(n__0, n__zeros)), activate(y0))) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)) ISNATLIST(n__cons(n__zeros, y0)) -> U51^1(isNat(n__cons(n__0, n__zeros)), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U51^1(tt, n__zeros) -> ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]: (U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)),U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros))) ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: U51^1(tt, n__length(x0)) -> ISNATLIST(length(x0)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ISNATLIST(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + x_2 POL(U51^1(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 1 + 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 1 + 2*x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: ISNATLIST(n__cons(n__length(x0), y1)) -> U51^1(isNat(length(x0)), activate(y1)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ISNATLIST(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U51^1(x_1, x_2)) = x_1 + 2*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 POL(U62(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 1 + x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 1 + x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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. ISNATLIST(n__cons(n__s(x0), y1)) -> U51^1(isNat(n__s(x0)), activate(y1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U51^1_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U21_1(x_1) ) = 2 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = x_1 POL( isNat_1(x_1) ) = 2 POL( n__0 ) = 0 POL( tt ) = 2 POL( n__s_1(x_1) ) = 2 POL( activate_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U52_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatList_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( n__nil ) = 2 POL( nil ) = 2 POL( ISNATLIST_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X 0 -> n__0 length(X) -> n__length(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) zeros -> cons(0, n__zeros) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(tt, x0) -> ISNATLIST(x0) ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(x0), activate(y1)) U51^1(tt, n__cons(x0, x1)) -> ISNATLIST(n__cons(x0, x1)) ISNATLIST(n__cons(n__0, y0)) -> U51^1(isNat(n__0), activate(y0)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(0, n__zeros)) U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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(isNat(n__0), activate(n__zeros)) evaluates to t =U51^1(isNat(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(isNat(n__0), activate(n__zeros)) -> U51^1(isNat(n__0), n__zeros) with rule activate(X) -> X at position [1] and matcher [X / n__zeros] U51^1(isNat(n__0), n__zeros) -> U51^1(tt, n__zeros) with rule isNat(n__0) -> tt at position [0] and matcher [ ] U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)) with rule U51^1(tt, n__zeros) -> ISNATLIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ] ISNATLIST(n__cons(n__0, n__zeros)) -> U51^1(isNat(n__0), activate(n__zeros)) with rule ISNATLIST(n__cons(x0, y1)) -> U51^1(isNat(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: U61^1(tt, L, N) -> U62^1(isNat(activate(N)), activate(L)) U62^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> U61^1(isNatList(activate(L)), activate(L), N) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (80) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U61^1(tt, L, N) -> U62^1(isNat(activate(N)), activate(L)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( LENGTH_1(x_1) ) = max{0, -2} POL( U61^1_3(x_1, ..., x_3) ) = x_1 POL( U62^1_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U61_3(x_1, ..., x_3) ) = 2x_1 POL( U62_2(x_1, x_2) ) = 2 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = max{0, 2x_1 - 2} POL( U21_1(x_1) ) = 2 POL( U52_1(x_1) ) = max{0, 2x_1 - 1} POL( length_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = x_1 + 2 POL( activate_1(x_1) ) = 0 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( n__s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( n__nil ) = 1 POL( nil ) = 2 POL( tt ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__length(X)) -> length(X) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(X) -> n__length(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) ---------------------------------------- (81) Obligation: Q DP problem: The TRS P consists of the following rules: U62^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> U61^1(isNatList(activate(L)), activate(L), N) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (82) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (83) TRUE ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(V1)) -> ISNAT(activate(V1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNAT(n__s(V1)) -> ISNAT(activate(V1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNAT_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 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( n__s_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = 2x_1 + 2 POL( n__cons_2(x_1, x_2) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( n__nil ) = 2 POL( nil ) = 2 POL( U61_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( isNat_1(x_1) ) = 2 POL( U21_1(x_1) ) = x_1 POL( U62_2(x_1, x_2) ) = 2x_1 + 2 POL( tt ) = 2 POL( U52_1(x_1) ) = max{0, 2x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X length(X) -> n__length(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (86) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U62(tt, L) -> s(length(activate(L))) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (88) YES ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, V2) -> ISNATILIST(activate(V2)) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNat(activate(V1)), activate(V2)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U21(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 0 POL(ISNATILIST(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U41^1(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = x_1 POL(n__nil) = 2 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 2 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, V2) -> ISNATILIST(activate(V2)) ISNATILIST(n__cons(V1, V2)) -> U41^1(isNat(activate(V1)), activate(V2)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, V2) -> ISNATILIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__zeros) -> ISNATILIST(zeros),U41^1(tt, n__zeros) -> ISNATILIST(zeros)) (U41^1(tt, n__0) -> ISNATILIST(0),U41^1(tt, n__0) -> ISNATILIST(0)) (U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)),U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0))) (U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)),U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0))) (U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)),U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1))) (U41^1(tt, n__nil) -> ISNATILIST(nil),U41^1(tt, n__nil) -> ISNATILIST(nil)) (U41^1(tt, x0) -> ISNATILIST(x0),U41^1(tt, x0) -> ISNATILIST(x0)) ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(V1, V2)) -> U41^1(isNat(activate(V1)), activate(V2)) U41^1(tt, n__zeros) -> ISNATILIST(zeros) U41^1(tt, n__0) -> ISNATILIST(0) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) U41^1(tt, n__nil) -> ISNATILIST(nil) U41^1(tt, x0) -> ISNATILIST(x0) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(V1, V2)) -> U41^1(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__zeros, y1)) -> U41^1(isNat(zeros), activate(y1)),ISNATILIST(n__cons(n__zeros, y1)) -> U41^1(isNat(zeros), activate(y1))) (ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1)),ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1))) (ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)),ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1))) (ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)),ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1))) (ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)),ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1))) (ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)),ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1))) (ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)),ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1))) ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__zeros) -> ISNATILIST(zeros) U41^1(tt, n__0) -> ISNATILIST(0) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) U41^1(tt, n__nil) -> ISNATILIST(nil) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(n__zeros, y1)) -> U41^1(isNat(zeros), activate(y1)) ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, n__zeros) -> ISNATILIST(zeros) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)),U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros))) (U41^1(tt, n__zeros) -> ISNATILIST(n__zeros),U41^1(tt, n__zeros) -> ISNATILIST(n__zeros)) ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__0) -> ISNATILIST(0) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) U41^1(tt, n__nil) -> ISNATILIST(nil) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(n__zeros, y1)) -> U41^1(isNat(zeros), activate(y1)) ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) U41^1(tt, n__zeros) -> ISNATILIST(n__zeros) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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: ISNATILIST(n__cons(n__zeros, y1)) -> U41^1(isNat(zeros), activate(y1)) U41^1(tt, n__0) -> ISNATILIST(0) ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y1)) -> U41^1(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)),ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0))) (ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__zeros), activate(y0)),ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__zeros), activate(y0))) ---------------------------------------- (101) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__0) -> ISNATILIST(0) ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__zeros), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (102) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (103) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1)) U41^1(tt, n__0) -> ISNATILIST(0) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (104) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__0, y1)) -> U41^1(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)),ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0))) ---------------------------------------- (105) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__0) -> ISNATILIST(0) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (106) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, n__0) -> ISNATILIST(0) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__0) -> ISNATILIST(n__0),U41^1(tt, n__0) -> ISNATILIST(n__0)) ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__0) -> ISNATILIST(n__0) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (108) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (109) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (110) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, n__s(x0)) -> ISNATILIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__s(x0)) -> ISNATILIST(n__s(x0)),U41^1(tt, n__s(x0)) -> ISNATILIST(n__s(x0))) ---------------------------------------- (111) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__s(x0)) -> ISNATILIST(n__s(x0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros 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(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (114) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)),ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1))) ---------------------------------------- (115) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)),U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1))) ---------------------------------------- (117) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> U41^1(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__cons(x0, x1), y2)) -> U41^1(isNat(n__cons(x0, x1)), activate(y2)),ISNATILIST(n__cons(n__cons(x0, x1), y2)) -> U41^1(isNat(n__cons(x0, x1)), activate(y2))) ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__cons(x0, x1), y2)) -> U41^1(isNat(n__cons(x0, x1)), activate(y2)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(nil) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, n__nil) -> ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__nil) -> ISNATILIST(n__nil),U41^1(tt, n__nil) -> ISNATILIST(n__nil)) ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__nil) -> ISNATILIST(n__nil) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (125) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__nil, y1)) -> U41^1(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__nil, y0)) -> U41^1(isNat(n__nil), activate(y0)),ISNATILIST(n__cons(n__nil, y0)) -> U41^1(isNat(n__nil), activate(y0))) ---------------------------------------- (127) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) ISNATILIST(n__cons(n__nil, y0)) -> U41^1(isNat(n__nil), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (128) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (129) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (130) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, n__zeros) -> ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)),U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros))) (U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)),U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros))) ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (132) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__cons(0, n__zeros)), activate(y0)),ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__cons(0, n__zeros)), activate(y0))) (ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(n__0, n__zeros)), activate(y0)),ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(n__0, n__zeros)), activate(y0))) ---------------------------------------- (133) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__cons(0, n__zeros)), activate(y0)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(n__0, n__zeros)), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (134) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(n__0, n__zeros)), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (136) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__cons(n__0, n__zeros)), activate(y0)),ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__cons(n__0, n__zeros)), activate(y0))) ---------------------------------------- (137) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)) ISNATILIST(n__cons(n__zeros, y0)) -> U41^1(isNat(n__cons(n__0, n__zeros)), activate(y0)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (138) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (139) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (140) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U41^1(tt, n__zeros) -> ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]: (U41^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)),U41^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros))) ---------------------------------------- (141) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (142) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: U41^1(tt, n__length(x0)) -> ISNATILIST(length(x0)) ISNATILIST(n__cons(n__length(x0), y1)) -> U41^1(isNat(length(x0)), activate(y1)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ISNATILIST(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U41^1(x_1, x_2)) = x_1 + 2*x_2 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2 + 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__length(x_1)) = 2 + 2*x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (143) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (144) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__s(x0), y1)) -> U41^1(isNat(n__s(x0)), activate(y1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U41^1_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U21_1(x_1) ) = 2 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = x_1 POL( isNat_1(x_1) ) = 2 POL( n__0 ) = 0 POL( tt ) = 2 POL( n__s_1(x_1) ) = 2 POL( activate_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U52_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatList_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( n__nil ) = 2 POL( nil ) = 2 POL( ISNATILIST_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X 0 -> n__0 length(X) -> n__length(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U21(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) zeros -> cons(0, n__zeros) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (145) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(tt, x0) -> ISNATILIST(x0) ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(x0), activate(y1)) U41^1(tt, n__cons(x0, x1)) -> ISNATILIST(n__cons(x0, x1)) ISNATILIST(n__cons(n__0, y0)) -> U41^1(isNat(n__0), activate(y0)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(0, n__zeros)) U41^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> U21(isNat(activate(V1))) U21(tt) -> tt nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) s(X) -> n__s(X) length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) length(X) -> n__length(X) isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt 0 -> n__0 zeros -> cons(0, n__zeros) zeros -> n__zeros Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (146) 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 = U41^1(isNat(n__0), activate(n__zeros)) evaluates to t =U41^1(isNat(n__0), activate(n__zeros)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence U41^1(isNat(n__0), activate(n__zeros)) -> U41^1(isNat(n__0), n__zeros) with rule activate(X) -> X at position [1] and matcher [X / n__zeros] U41^1(isNat(n__0), n__zeros) -> U41^1(tt, n__zeros) with rule isNat(n__0) -> tt at position [0] and matcher [ ] U41^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)) with rule U41^1(tt, n__zeros) -> ISNATILIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ] ISNATILIST(n__cons(n__0, n__zeros)) -> U41^1(isNat(n__0), activate(n__zeros)) with rule ISNATILIST(n__cons(x0, y1)) -> U41^1(isNat(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. ---------------------------------------- (147) NO