NO proof of Transformed_CSR_04_LengthOfFiniteLists_nokinds_Z.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 71 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 12 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) MRRProof [EQUIVALENT, 0 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) QDPOrderProof [EQUIVALENT, 15 ms] (13) QDP (14) DependencyGraphProof [EQUIVALENT, 0 ms] (15) AND (16) QDP (17) QDPOrderProof [EQUIVALENT, 0 ms] (18) QDP (19) PisEmptyProof [EQUIVALENT, 0 ms] (20) YES (21) QDP (22) TransformationProof [EQUIVALENT, 0 ms] (23) QDP (24) DependencyGraphProof [EQUIVALENT, 0 ms] (25) QDP (26) TransformationProof [EQUIVALENT, 0 ms] (27) QDP (28) DependencyGraphProof [EQUIVALENT, 0 ms] (29) QDP (30) TransformationProof [EQUIVALENT, 0 ms] (31) QDP (32) DependencyGraphProof [EQUIVALENT, 0 ms] (33) AND (34) QDP (35) MRRProof [EQUIVALENT, 0 ms] (36) QDP (37) MRRProof [EQUIVALENT, 10 ms] (38) QDP (39) QDPOrderProof [EQUIVALENT, 87 ms] (40) QDP (41) QDPOrderProof [EQUIVALENT, 19 ms] (42) QDP (43) QDPOrderProof [EQUIVALENT, 23 ms] (44) QDP (45) QDPOrderProof [EQUIVALENT, 51 ms] (46) QDP (47) NonTerminationLoopProof [COMPLETE, 1615 ms] (48) NO (49) QDP (50) MRRProof [EQUIVALENT, 0 ms] (51) QDP (52) MRRProof [EQUIVALENT, 0 ms] (53) QDP (54) MRRProof [EQUIVALENT, 17 ms] (55) QDP (56) QDPOrderProof [EQUIVALENT, 53 ms] (57) QDP (58) QDPOrderProof [EQUIVALENT, 44 ms] (59) QDP (60) QDPOrderProof [EQUIVALENT, 14 ms] (61) QDP (62) QDP (63) QDPOrderProof [EQUIVALENT, 57 ms] (64) QDP (65) DependencyGraphProof [EQUIVALENT, 0 ms] (66) TRUE ---------------------------------------- (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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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)) = x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = 2 + 2*x_1 POL(isNatList(x_1)) = 2*x_1 POL(length(x_1)) = 2 + 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)) = 2 + 2*x_1 POL(n__isNatList(x_1)) = 2*x_1 POL(n__length(x_1)) = 2 + 2*x_1 POL(n__nil) = 2 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 2 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 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)) 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__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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) 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(X) ACTIVATE(n__s(X)) -> S(X) ACTIVATE(n__cons(X1, X2)) -> CONS(X1, X2) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__length(X)) -> LENGTH(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__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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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: ACTIVATE(n__length(X)) -> LENGTH(X) 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(LENGTH(x_1)) = 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 + x_2 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = 2 + 2*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)) = 2 + 2*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 ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 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__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) 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) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) 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))) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (12) 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) 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 + 1 POL( ISNAT_1(x_1) ) = x_1 + 2 POL( n__isNatIList_1(x_1) ) = 2x_1 + 1 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) ) = 2x_1 POL( length_1(x_1) ) = 2x_1 POL( n__s_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( isNat_1(x_1) ) = 2x_1 + 2 POL( tt ) = 2 POL( isNatList_1(x_1) ) = x_1 + 1 POL( n__nil ) = 0 POL( nil ) = 0 POL( n__isNat_1(x_1) ) = 2x_1 + 2 POL( U11_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( ACTIVATE_1(x_1) ) = x_1 + 1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( ISNATLIST_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__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) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatList(X) -> n__isNatList(X) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (13) 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))) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (15) Complex Obligation (AND) ---------------------------------------- (16) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (17) 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) ) = 2x_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 + 1 POL( s_1(x_1) ) = 2x_1 + 1 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( n__isNatIList_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = 2 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( isNat_1(x_1) ) = 1 POL( tt ) = 1 POL( n__isNatList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( n__nil ) = 0 POL( nil ) = 0 POL( n__isNat_1(x_1) ) = 1 POL( U11_2(x_1, x_2) ) = max{0, 2x_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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatList(X) -> n__isNatList(X) isNatIList(X) -> n__isNatIList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) isNat(n__s(V1)) -> isNat(activate(V1)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (18) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (22) 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(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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)))) ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(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(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (24) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (25) 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))) 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(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (26) 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(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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)))) ---------------------------------------- (27) 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))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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))) 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(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (29) 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(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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))) ---------------------------------------- (31) 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))) 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(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (32) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (33) Complex Obligation (AND) ---------------------------------------- (34) 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(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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(x0)), n__isNatList(activate(y1))) 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 + 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)) = 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) = 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 ---------------------------------------- (36) 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(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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 + 2*x_1 POL(AND(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(ISNATLIST(x_1)) = 2 + 2*x_1 POL(U11(x_1, x_2)) = x_1 + 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)) = 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)) = 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 ---------------------------------------- (38) 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(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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(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( zeros ) = 1 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 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 + 1 POL( n__isNat_1(x_1) ) = 0 POL( n__length_1(x_1) ) = x_1 + 1 POL( length_1(x_1) ) = x_1 + 2 POL( s_1(x_1) ) = 0 POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( n__isNatIList_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = 1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( isNatList_1(x_1) ) = x_1 + 1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 2x_2 + 1 POL( ISNATLIST_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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (40) 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(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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(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) ) = x_1 + x_2 + 2 POL( n__isNatList_1(x_1) ) = 2x_1 POL( zeros ) = 2 POL( cons_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = 0 POL( n__0 ) = 0 POL( tt ) = 0 POL( n__s_1(x_1) ) = 2 POL( activate_1(x_1) ) = x_1 + 2 POL( n__isNat_1(x_1) ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = x_1 + x_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_1 + x_2 + 2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__nil ) = 0 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = 2 POL( ISNATLIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) isNatList(X) -> n__isNatList(X) length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (42) 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__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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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))) 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 + x_2 - 2} POL( n__isNatList_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = 2x_1 + 2 POL( n__0 ) = 0 POL( tt ) = 2 POL( n__s_1(x_1) ) = x_1 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 2x_1 + 2 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( n__isNatIList_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 1 POL( and_2(x_1, x_2) ) = x_2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 0 POL( ISNATLIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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 isNatList(X) -> n__isNatList(X) length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (44) 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(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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 - 1} POL( n__isNatList_1(x_1) ) = 2x_1 + 1 POL( zeros ) = 2 POL( cons_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( 0 ) = 0 POL( n__zeros ) = 1 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 + 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) ) = x_1 + x_2 + 1 POL( n__isNatIList_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 2 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1} POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__nil ) = 0 POL( nil ) = 1 POL( U11_2(x_1, x_2) ) = 2 POL( ISNATLIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatList(X) -> n__isNatList(X) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (46) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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. ---------------------------------------- (48) NO ---------------------------------------- (49) 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__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(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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: ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(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)) = x_1 POL(U11(x_1, x_2)) = 2 + x_1 + 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) = 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 ---------------------------------------- (51) 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__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__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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: 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)) = 1 + 2*x_1 POL(AND(x_1, x_2)) = 1 + x_1 + 2*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)) = 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)) = 2*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) = 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 ---------------------------------------- (53) 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__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__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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: 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)) = 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)) = 2*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)) = 2*x_1 POL(n__length(x_1)) = 2*x_1 POL(n__nil) = 2 POL(n__s(x_1)) = x_1 POL(n__zeros) = 0 POL(nil) = 2 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (55) 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__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__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), 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))) 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 - 2} POL( n__isNatIList_1(x_1) ) = 2x_1 + 1 POL( zeros ) = 1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 1 POL( isNat_1(x_1) ) = 2 POL( n__0 ) = 0 POL( tt ) = 2 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 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) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( and_2(x_1, x_2) ) = x_2 POL( n__isNatList_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 2x_1 + 1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( ACTIVATE_1(x_1) ) = x_1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) nil -> n__nil ---------------------------------------- (57) 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__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(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) ) = x_2 + 2 POL( n__isNatIList_1(x_1) ) = 2x_1 POL( 0 ) = 0 POL( n__0 ) = 0 POL( isNat_1(x_1) ) = 2 POL( tt ) = 2 POL( n__s_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 + 2 POL( n__isNat_1(x_1) ) = 1 POL( n__zeros ) = 0 POL( zeros ) = 2 POL( n__length_1(x_1) ) = 2x_1 POL( length_1(x_1) ) = 2x_1 + 1 POL( s_1(x_1) ) = max{0, -1} POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( n__isNatList_1(x_1) ) = max{0, x_1 - 2} POL( isNatList_1(x_1) ) = x_1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( ACTIVATE_1(x_1) ) = x_1 + 2 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatList(X) -> n__isNatList(X) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (59) 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__s(x0), y1)) -> AND(isNat(s(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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(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) ) = x_1 + x_2 POL( n__isNatIList_1(x_1) ) = 2x_1 POL( 0 ) = 0 POL( n__0 ) = 0 POL( isNat_1(x_1) ) = 2 POL( tt ) = 2 POL( n__s_1(x_1) ) = 2 POL( activate_1(x_1) ) = x_1 + 2 POL( n__isNat_1(x_1) ) = 2 POL( n__zeros ) = 0 POL( zeros ) = 2 POL( n__length_1(x_1) ) = 2x_1 POL( length_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = 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( isNatIList_1(x_1) ) = 2x_1 + 2 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( n__isNatList_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 2 POL( n__nil ) = 0 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( ACTIVATE_1(x_1) ) = x_1 + 2 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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 s(X) -> n__s(X) length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatList(X) -> n__isNatList(X) isNatIList(X) -> n__isNatIList(X) U11(tt, L) -> s(length(activate(L))) zeros -> cons(0, n__zeros) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (61) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (63) 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)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( LENGTH_1(x_1) ) = max{0, -2} POL( U11^1_2(x_1, x_2) ) = x_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) ) = 1 POL( s_1(x_1) ) = 1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( n__isNatIList_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( and_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( isNat_1(x_1) ) = 1 POL( tt ) = 1 POL( n__isNatList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = max{0, -2} POL( n__nil ) = 0 POL( nil ) = 0 POL( n__isNat_1(x_1) ) = 1 POL( U11_2(x_1, x_2) ) = 2x_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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) isNatIList(X) -> n__isNatIList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) isNat(n__s(V1)) -> isNat(activate(V1)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: 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(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(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. ---------------------------------------- (65) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (66) TRUE