/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- NO proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 128 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 32 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 29 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) MRRProof [EQUIVALENT, 60 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) QDPOrderProof [EQUIVALENT, 47 ms] (15) QDP (16) PisEmptyProof [EQUIVALENT, 0 ms] (17) YES (18) QDP (19) MRRProof [EQUIVALENT, 24 ms] (20) QDP (21) QDPOrderProof [EQUIVALENT, 68 ms] (22) QDP (23) QDPOrderProof [EQUIVALENT, 45 ms] (24) QDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) AND (27) QDP (28) QDPOrderProof [EQUIVALENT, 25 ms] (29) QDP (30) PisEmptyProof [EQUIVALENT, 0 ms] (31) YES (32) QDP (33) QDPOrderProof [EQUIVALENT, 58 ms] (34) QDP (35) QDPOrderProof [EQUIVALENT, 21 ms] (36) QDP (37) TransformationProof [EQUIVALENT, 0 ms] (38) QDP (39) DependencyGraphProof [EQUIVALENT, 0 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) DependencyGraphProof [EQUIVALENT, 0 ms] (44) QDP (45) TransformationProof [EQUIVALENT, 0 ms] (46) QDP (47) DependencyGraphProof [EQUIVALENT, 0 ms] (48) AND (49) QDP (50) MRRProof [EQUIVALENT, 23 ms] (51) QDP (52) MRRProof [EQUIVALENT, 22 ms] (53) QDP (54) MRRProof [EQUIVALENT, 29 ms] (55) QDP (56) QDPOrderProof [EQUIVALENT, 40 ms] (57) QDP (58) QDPOrderProof [EQUIVALENT, 27 ms] (59) QDP (60) QDPOrderProof [EQUIVALENT, 84 ms] (61) QDP (62) QDPOrderProof [EQUIVALENT, 23 ms] (63) QDP (64) NonTerminationLoopProof [COMPLETE, 1204 ms] (65) NO (66) QDP (67) MRRProof [EQUIVALENT, 19 ms] (68) QDP (69) MRRProof [EQUIVALENT, 20 ms] (70) QDP (71) MRRProof [EQUIVALENT, 19 ms] (72) QDP (73) QDPOrderProof [EQUIVALENT, 24 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 88 ms] (76) QDP (77) QDPOrderProof [EQUIVALENT, 107 ms] (78) QDP (79) QDPOrderProof [EQUIVALENT, 19 ms] (80) QDP (81) QDPOrderProof [EQUIVALENT, 17 ms] (82) QDP (83) NonTerminationLoopProof [COMPLETE, 1142 ms] (84) NO ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = 2*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__isNat(x_1)) = 2*x_1 POL(n__isNatIList(x_1)) = 1 + 2*x_1 POL(n__isNatList(x_1)) = x_1 POL(n__length(x_1)) = 2*x_1 POL(n__nil) = 1 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 1 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) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt 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, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + 2*x_2 POL(cons(x_1, x_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)) = 1 + 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = x_1 + 2*x_2 POL(n__isNat(x_1)) = x_1 POL(n__isNatIList(x_1)) = x_1 POL(n__isNatList(x_1)) = x_1 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 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isNat(n__length(V1)) -> isNatList(activate(V1)) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: ZEROS -> CONS(0, n__zeros) ZEROS -> 0^1 U11^1(tt, L) -> S(length(activate(L))) U11^1(tt, L) -> LENGTH(activate(L)) U11^1(tt, L) -> ACTIVATE(L) AND(tt, X) -> ACTIVATE(X) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(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)) -> AND(isNat(activate(V1)), n__isNatList(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)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) LENGTH(cons(N, L)) -> AND(isNatList(activate(L)), n__isNat(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(activate(X)) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> S(activate(X)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> CONS(activate(X1), X2) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__nil) -> NIL ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ACTIVATE(n__isNat(X)) -> ISNAT(X) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__length(X)) -> LENGTH(activate(X)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> AND(isNatList(activate(L)), n__isNat(N)) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) LENGTH(cons(N, L)) -> ACTIVATE(L) U11^1(tt, L) -> ACTIVATE(L) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) 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)) -> AND(isNatList(activate(L)), n__isNat(N)) ACTIVATE(n__length(X)) -> ACTIVATE(X) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) LENGTH(cons(N, L)) -> ACTIVATE(L) U11^1(tt, L) -> ACTIVATE(L) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = 1 + x_1 POL(AND(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(ISNAT(x_1)) = 1 + 2*x_1 POL(ISNATILIST(x_1)) = 1 + 2*x_1 POL(ISNATLIST(x_1)) = 1 + 2*x_1 POL(LENGTH(x_1)) = 2 + 2*x_1 POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U11^1(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + 2*x_2 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)) = 2*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__isNat(x_1)) = 2*x_1 POL(n__isNatIList(x_1)) = 2*x_1 POL(n__isNatList(x_1)) = 2*x_1 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 ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__length(X)) -> LENGTH(activate(X)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, L) -> LENGTH(activate(L)) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) 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 1 less node. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U11^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(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) ) = 2 POL( U11^1_2(x_1, x_2) ) = x_1 + 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( n__s_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( isNat_1(x_1) ) = 2 POL( tt ) = 2 POL( n__isNatList_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = max{0, -2} POL( n__nil ) = 2 POL( nil ) = 2 POL( n__isNat_1(x_1) ) = 2 POL( U11_2(x_1, x_2) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X isNatList(X) -> n__isNatList(X) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) isNat(n__s(V1)) -> isNat(activate(V1)) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (15) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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: ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = x_1 POL(AND(x_1, x_2)) = x_1 + x_2 POL(ISNAT(x_1)) = x_1 POL(ISNATILIST(x_1)) = 2 + 2*x_1 POL(ISNATLIST(x_1)) = x_1 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = 2*x_1 POL(isNatIList(x_1)) = 2 + 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__isNat(x_1)) = 2*x_1 POL(n__isNatIList(x_1)) = 2 + 2*x_1 POL(n__isNatList(x_1)) = x_1 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 ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = x_2 POL( ISNAT_1(x_1) ) = 2x_1 POL( n__isNatIList_1(x_1) ) = 0 POL( n__isNatList_1(x_1) ) = x_1 + 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = 0 POL( n__s_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 2x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = 2x_1 POL( tt ) = 0 POL( isNatList_1(x_1) ) = x_1 + 1 POL( n__nil ) = 2 POL( nil ) = 2 POL( n__isNat_1(x_1) ) = 2x_1 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ACTIVATE_1(x_1) ) = x_1 POL( ISNATILIST_1(x_1) ) = 0 POL( ISNATLIST_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__isNat(X)) -> ISNAT(X) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__isNat(X)) -> ISNAT(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = 2x_2 POL( ISNAT_1(x_1) ) = 2x_1 + 1 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( n__isNatList_1(x_1) ) = max{0, -2} POL( activate_1(x_1) ) = 2x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -2} POL( n__s_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 2x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 POL( cons_2(x_1, x_2) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = 2x_2 POL( isNat_1(x_1) ) = 2x_1 + 2 POL( tt ) = 0 POL( isNatList_1(x_1) ) = max{0, -2} POL( n__nil ) = 1 POL( nil ) = 2 POL( n__isNat_1(x_1) ) = 2x_1 + 1 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ACTIVATE_1(x_1) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 0 POL( ISNATLIST_1(x_1) ) = 0 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(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (26) Complex Obligation (AND) ---------------------------------------- (27) 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: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) 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) ) = x_1 + 1 POL( activate_1(x_1) ) = 2x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -2} POL( n__s_1(x_1) ) = 2x_1 + 1 POL( s_1(x_1) ) = 2x_1 + 1 POL( n__cons_2(x_1, x_2) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( isNat_1(x_1) ) = 2 POL( tt ) = 2 POL( n__isNatList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = max{0, -2} POL( n__nil ) = 1 POL( nil ) = 2 POL( n__isNat_1(x_1) ) = 1 POL( U11_2(x_1, x_2) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) isNatList(X) -> n__isNatList(X) isNat(n__s(V1)) -> isNat(activate(V1)) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (29) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( n__isNatList_1(x_1) ) = max{0, -2} POL( activate_1(x_1) ) = 2x_1 POL( n__zeros ) = 1 POL( zeros ) = 1 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -2} POL( n__s_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 2x_1 POL( n__cons_2(x_1, x_2) ) = x_1 + 1 POL( cons_2(x_1, x_2) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( isNat_1(x_1) ) = 1 POL( tt ) = 1 POL( isNatList_1(x_1) ) = max{0, -2} POL( n__nil ) = 1 POL( nil ) = 2 POL( n__isNat_1(x_1) ) = 1 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ACTIVATE_1(x_1) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 0 POL( ISNATLIST_1(x_1) ) = 0 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(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVATE(n__s(X)) -> ACTIVATE(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( n__isNatIList_1(x_1) ) = 0 POL( n__isNatList_1(x_1) ) = max{0, -2} POL( activate_1(x_1) ) = 2x_1 POL( n__zeros ) = 1 POL( zeros ) = 1 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -2} POL( n__s_1(x_1) ) = 2x_1 + 1 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( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( isNat_1(x_1) ) = 1 POL( tt ) = 1 POL( isNatList_1(x_1) ) = max{0, -2} POL( n__nil ) = 1 POL( nil ) = 2 POL( n__isNat_1(x_1) ) = 1 POL( U11_2(x_1, x_2) ) = 2x_1 POL( ACTIVATE_1(x_1) ) = 2x_1 + 2 POL( ISNATILIST_1(x_1) ) = 2 POL( ISNATLIST_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))),ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1)))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNATILIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))),ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1)))) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule AND(tt, X) -> ACTIVATE(X) we obtained the following new rules [LPAR04]: (AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)),AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3))) (AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)),AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (48) Complex Obligation (AND) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) 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__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = 2*x_1 POL(AND(x_1, x_2)) = 2*x_1 + 2*x_2 POL(ISNATLIST(x_1)) = 2*x_1 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + 2*x_2 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)) = 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__isNat(x_1)) = x_1 POL(n__isNatIList(x_1)) = x_1 POL(n__isNatList(x_1)) = x_1 POL(n__length(x_1)) = 2*x_1 POL(n__nil) = 1 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 1 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) 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__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = 2*x_1 POL(AND(x_1, x_2)) = x_1 + 2*x_2 POL(ISNATLIST(x_1)) = 2*x_1 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_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)) = x_1 + 2*x_2 POL(n__isNat(x_1)) = x_1 POL(n__isNatIList(x_1)) = 1 + 2*x_1 POL(n__isNatList(x_1)) = x_1 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 ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) 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)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = 2*x_1 POL(AND(x_1, x_2)) = 2*x_1 + 2*x_2 POL(ISNATLIST(x_1)) = 2*x_1 POL(U11(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + 2*x_2 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)) = 2 + 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__isNat(x_1)) = x_1 POL(n__isNatIList(x_1)) = x_1 POL(n__isNatList(x_1)) = x_1 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 ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) 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)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( n__isNatList_1(x_1) ) = x_1 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 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( n__isNat_1(x_1) ) = 2 POL( n__length_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( isNatList_1(x_1) ) = x_1 POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = 2 POL( ISNATLIST_1(x_1) ) = x_1 POL( ACTIVATE_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatList(X) -> n__isNatList(X) length(X) -> n__length(X) isNatIList(X) -> n__isNatIList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (58) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, x_2 - 2} POL( n__isNatList_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = max{0, -2} POL( n__0 ) = 0 POL( tt ) = 0 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 0 POL( n__length_1(x_1) ) = x_1 POL( length_1(x_1) ) = x_1 POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = x_2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ISNATLIST_1(x_1) ) = 2x_1 POL( ACTIVATE_1(x_1) ) = max{0, x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) isNatList(X) -> n__isNatList(X) length(X) -> n__length(X) s(X) -> n__s(X) isNatIList(X) -> n__isNatIList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (60) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( n__isNatList_1(x_1) ) = 2x_1 POL( zeros ) = 2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 2 POL( isNat_1(x_1) ) = 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( n__s_1(x_1) ) = 2 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 1 POL( n__length_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( isNatList_1(x_1) ) = 2x_1 POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = 2 POL( ISNATLIST_1(x_1) ) = 2x_1 POL( ACTIVATE_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) length(X) -> n__length(X) s(X) -> n__s(X) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (62) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( n__isNatList_1(x_1) ) = x_1 POL( 0 ) = 0 POL( n__0 ) = 0 POL( isNat_1(x_1) ) = 1 POL( tt ) = 1 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = 2x_1 + 2 POL( n__isNat_1(x_1) ) = 1 POL( n__zeros ) = 0 POL( zeros ) = 2 POL( n__length_1(x_1) ) = 1 POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2 POL( and_2(x_1, x_2) ) = 2x_2 + 2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__nil ) = 1 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ACTIVATE_1(x_1) ) = 2x_1 POL( ISNATLIST_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0 -> n__0 isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X cons(X1, X2) -> n__cons(X1, X2) length(X) -> n__length(X) s(X) -> n__s(X) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) 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 = ACTIVATE(n__isNatList(activate(n__zeros))) evaluates to t =ACTIVATE(n__isNatList(activate(n__zeros))) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence ACTIVATE(n__isNatList(activate(n__zeros))) -> ACTIVATE(n__isNatList(zeros)) with rule activate(n__zeros) -> zeros at position [0,0] and matcher [ ] ACTIVATE(n__isNatList(zeros)) -> ACTIVATE(n__isNatList(cons(0, n__zeros))) with rule zeros -> cons(0, n__zeros) at position [0,0] and matcher [ ] ACTIVATE(n__isNatList(cons(0, n__zeros))) -> ACTIVATE(n__isNatList(cons(n__0, n__zeros))) with rule 0 -> n__0 at position [0,0,0] and matcher [ ] ACTIVATE(n__isNatList(cons(n__0, n__zeros))) -> ACTIVATE(n__isNatList(n__cons(n__0, n__zeros))) with rule cons(X1, X2) -> n__cons(X1, X2) at position [0,0] and matcher [X1 / n__0, X2 / n__zeros] ACTIVATE(n__isNatList(n__cons(n__0, n__zeros))) -> ISNATLIST(n__cons(n__0, n__zeros)) with rule ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) at position [] and matcher [X / n__cons(n__0, n__zeros)] ISNATLIST(n__cons(n__0, n__zeros)) -> AND(isNat(n__0), n__isNatList(activate(n__zeros))) with rule ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) at position [] and matcher [x0 / n__0, y1 / n__zeros] AND(isNat(n__0), n__isNatList(activate(n__zeros))) -> AND(tt, n__isNatList(activate(n__zeros))) with rule isNat(n__0) -> tt at position [0] and matcher [ ] AND(tt, n__isNatList(activate(n__zeros))) -> ACTIVATE(n__isNatList(activate(n__zeros))) with rule AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) 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. ---------------------------------------- (65) NO ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) 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: ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = 2*x_1 POL(AND(x_1, x_2)) = x_1 + 2*x_2 POL(ISNATILIST(x_1)) = 2*x_1 POL(U11(x_1, x_2)) = x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_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)) = 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = x_1 + 2*x_2 POL(n__isNat(x_1)) = x_1 POL(n__isNatIList(x_1)) = x_1 POL(n__isNatList(x_1)) = x_1 POL(n__length(x_1)) = 2*x_1 POL(n__nil) = 1 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 1 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) 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: ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = 2*x_1 POL(AND(x_1, x_2)) = 2*x_1 + 2*x_2 POL(ISNATILIST(x_1)) = 2*x_1 POL(U11(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + 2*x_2 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)) = 1 + 2*x_1 POL(n__0) = 0 POL(n__cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__isNat(x_1)) = x_1 POL(n__isNatIList(x_1)) = x_1 POL(n__isNatList(x_1)) = x_1 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 ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X 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: ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = x_1 POL(AND(x_1, x_2)) = x_1 + x_2 POL(ISNATILIST(x_1)) = 1 + 2*x_1 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = 2*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__isNat(x_1)) = 2*x_1 POL(n__isNatIList(x_1)) = 1 + 2*x_1 POL(n__isNatList(x_1)) = x_1 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 ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( n__isNatIList_1(x_1) ) = x_1 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 1 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( isNatIList_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( n__isNatList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = 0 POL( ISNATILIST_1(x_1) ) = 2x_1 POL( ACTIVATE_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatList(X) -> n__isNatList(X) length(X) -> n__length(X) isNatIList(X) -> n__isNatIList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( n__isNatIList_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 1 POL( n__length_1(x_1) ) = 1 POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( n__isNatList_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = 1 POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_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: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatList(X) -> n__isNatList(X) length(X) -> n__length(X) isNatIList(X) -> n__isNatIList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( n__isNatIList_1(x_1) ) = 2x_1 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( n__s_1(x_1) ) = 2 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 1 POL( n__length_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = 2x_1 POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( n__isNatList_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = max{0, -2} POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 2x_1 POL( ACTIVATE_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) length(X) -> n__length(X) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( n__isNatIList_1(x_1) ) = x_1 POL( zeros ) = 2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 2 POL( isNat_1(x_1) ) = 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 1 POL( n__length_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( n__isNatList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = x_1 POL( ISNATILIST_1(x_1) ) = 2x_1 POL( ACTIVATE_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) length(X) -> n__length(X) s(X) -> n__s(X) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = 2x_2 POL( n__isNatIList_1(x_1) ) = x_1 POL( 0 ) = 0 POL( n__0 ) = 0 POL( isNat_1(x_1) ) = 1 POL( tt ) = 1 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 + 1 POL( n__isNat_1(x_1) ) = 1 POL( n__zeros ) = 1 POL( zeros ) = 2 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( isNatIList_1(x_1) ) = x_1 + 1 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1} POL( n__isNatList_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 1 POL( n__nil ) = 1 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ACTIVATE_1(x_1) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0 -> n__0 isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) activate(n__nil) -> nil activate(n__isNat(X)) -> isNat(X) activate(X) -> X cons(X1, X2) -> n__cons(X1, X2) length(X) -> n__length(X) s(X) -> n__s(X) isNatIList(X) -> n__isNatIList(X) isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = ACTIVATE(n__isNatIList(activate(n__zeros))) evaluates to t =ACTIVATE(n__isNatIList(activate(n__zeros))) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence ACTIVATE(n__isNatIList(activate(n__zeros))) -> ACTIVATE(n__isNatIList(zeros)) with rule activate(n__zeros) -> zeros at position [0,0] and matcher [ ] ACTIVATE(n__isNatIList(zeros)) -> ACTIVATE(n__isNatIList(cons(0, n__zeros))) with rule zeros -> cons(0, n__zeros) at position [0,0] and matcher [ ] ACTIVATE(n__isNatIList(cons(0, n__zeros))) -> ACTIVATE(n__isNatIList(cons(n__0, n__zeros))) with rule 0 -> n__0 at position [0,0,0] and matcher [ ] ACTIVATE(n__isNatIList(cons(n__0, n__zeros))) -> ACTIVATE(n__isNatIList(n__cons(n__0, n__zeros))) with rule cons(X1, X2) -> n__cons(X1, X2) at position [0,0] and matcher [X1 / n__0, X2 / n__zeros] ACTIVATE(n__isNatIList(n__cons(n__0, n__zeros))) -> ISNATILIST(n__cons(n__0, n__zeros)) with rule ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) at position [] and matcher [X / n__cons(n__0, n__zeros)] ISNATILIST(n__cons(n__0, n__zeros)) -> AND(isNat(n__0), n__isNatIList(activate(n__zeros))) with rule ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) at position [] and matcher [x0 / n__0, y1 / n__zeros] AND(isNat(n__0), n__isNatIList(activate(n__zeros))) -> AND(tt, n__isNatIList(activate(n__zeros))) with rule isNat(n__0) -> tt at position [0] and matcher [ ] AND(tt, n__isNatIList(activate(n__zeros))) -> ACTIVATE(n__isNatIList(activate(n__zeros))) with rule AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (84) NO