/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 74 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] (9) YES (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) QDPOrderProof [EQUIVALENT, 374 ms] (18) QDP (19) QDPOrderProof [EQUIVALENT, 347 ms] (20) QDP (21) DependencyGraphProof [EQUIVALENT, 0 ms] (22) AND (23) QDP (24) QDPOrderProof [EQUIVALENT, 304 ms] (25) QDP (26) DependencyGraphProof [EQUIVALENT, 0 ms] (27) TRUE (28) QDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) QDP (31) QDPSizeChangeProof [EQUIVALENT, 0 ms] (32) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) A__U11(tt, V1) -> A__ISNATILISTKIND(V1) A__U12(tt, V1) -> A__U13(a__isNatList(V1)) A__U12(tt, V1) -> A__ISNATLIST(V1) A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) A__U21(tt, V1) -> A__ISNATKIND(V1) A__U22(tt, V1) -> A__U23(a__isNat(V1)) A__U22(tt, V1) -> A__ISNAT(V1) A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) A__U31(tt, V) -> A__ISNATILISTKIND(V) A__U32(tt, V) -> A__U33(a__isNatList(V)) A__U32(tt, V) -> A__ISNATLIST(V) A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) A__U41(tt, V1, V2) -> A__ISNATKIND(V1) A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) A__U42(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) A__U43(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) A__U44(tt, V1, V2) -> A__ISNAT(V1) A__U45(tt, V2) -> A__U46(a__isNatIList(V2)) A__U45(tt, V2) -> A__ISNATILIST(V2) A__U51(tt, V2) -> A__U52(a__isNatIListKind(V2)) A__U51(tt, V2) -> A__ISNATILISTKIND(V2) A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) A__U81(tt, V1, V2) -> A__ISNATKIND(V1) A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) A__U82(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) A__U83(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) A__U84(tt, V1, V2) -> A__ISNAT(V1) A__U85(tt, V2) -> A__U86(a__isNatList(V2)) A__U85(tt, V2) -> A__ISNATLIST(V2) A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) A__U91(tt, L, N) -> A__ISNATILISTKIND(L) A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) A__U92(tt, L, N) -> A__ISNAT(N) A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) A__U93(tt, L, N) -> A__ISNATKIND(N) A__U94(tt, L) -> A__LENGTH(mark(L)) A__U94(tt, L) -> MARK(L) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) A__ISNATILIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__U61(a__isNatIListKind(V1)) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__U71(a__isNatKind(V1)) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) MARK(zeros) -> A__ZEROS MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> A__U12(mark(X1), X2) MARK(U12(X1, X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(U13(X)) -> A__U13(mark(X)) MARK(U13(X)) -> MARK(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> A__U22(mark(X1), X2) MARK(U22(X1, X2)) -> MARK(X1) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(U23(X)) -> A__U23(mark(X)) MARK(U23(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> A__U32(mark(X1), X2) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> A__U33(mark(X)) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> A__U42(mark(X1), X2, X3) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> A__U43(mark(X1), X2, X3) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> A__U44(mark(X1), X2, X3) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> A__U45(mark(X1), X2) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> A__U46(mark(X)) MARK(U46(X)) -> MARK(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(U51(X1, X2)) -> A__U51(mark(X1), X2) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> A__U52(mark(X)) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> A__U61(mark(X)) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> A__U71(mark(X)) MARK(U71(X)) -> MARK(X) MARK(U81(X1, X2, X3)) -> A__U81(mark(X1), X2, X3) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> A__U82(mark(X1), X2, X3) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> A__U83(mark(X1), X2, X3) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> A__U84(mark(X1), X2, X3) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(U85(X1, X2)) -> A__U85(mark(X1), X2) MARK(U85(X1, X2)) -> MARK(X1) MARK(U86(X)) -> A__U86(mark(X)) MARK(U86(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2)) -> A__U94(mark(X1), X2) MARK(U94(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 61 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: A__U51(tt, V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (6) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: A__U51(tt, V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) The TRS R consists of the following rules: a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) The graph contains the following edges 1 > 2 *A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) The graph contains the following edges 1 > 1 *A__U51(tt, V2) -> A__ISNATILISTKIND(V2) The graph contains the following edges 2 >= 1 *A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) The graph contains the following edges 1 > 1 *A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) The graph contains the following edges 1 > 1 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: A__U12(tt, V1) -> A__ISNATLIST(V1) A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) A__U85(tt, V2) -> A__ISNATLIST(V2) A__U84(tt, V1, V2) -> A__ISNAT(V1) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) A__U22(tt, V1) -> A__ISNAT(V1) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) The graph contains the following edges 1 > 2, 1 > 3 *A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) The graph contains the following edges 2 >= 2 *A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U85(tt, V2) -> A__ISNATLIST(V2) The graph contains the following edges 2 >= 1 *A__U12(tt, V1) -> A__ISNATLIST(V1) The graph contains the following edges 2 >= 1 *A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) The graph contains the following edges 3 >= 2 *A__U84(tt, V1, V2) -> A__ISNAT(V1) The graph contains the following edges 2 >= 1 *A__U22(tt, V1) -> A__ISNAT(V1) The graph contains the following edges 2 >= 1 *A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) The graph contains the following edges 2 >= 2 *A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) The graph contains the following edges 1 > 2 *A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) The graph contains the following edges 1 > 2 ---------------------------------------- (12) YES ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) A__U45(tt, V2) -> A__ISNATILIST(V2) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *A__U45(tt, V2) -> A__ISNATILIST(V2) The graph contains the following edges 2 >= 1 *A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) The graph contains the following edges 1 > 2, 1 > 3 *A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) The graph contains the following edges 3 >= 2 *A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(U85(X1, X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) A__U94(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) A__U94(tt, L) -> MARK(L) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2)) -> A__U94(mark(X1), X2) MARK(U94(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(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. MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2)) -> A__U94(mark(X1), X2) MARK(U94(X1, X2)) -> MARK(X1) MARK(cons(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__LENGTH_1(x_1) ) = max{0, x_1 - 1} POL( A__U91_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 1 POL( A__U92_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 1 POL( A__U93_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( A__U94_2(x_1, x_2) ) = 2x_2 + 1 POL( mark_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( a__zeros ) = 2 POL( U11_2(x_1, x_2) ) = x_1 POL( a__U11_2(x_1, x_2) ) = x_1 POL( U12_2(x_1, x_2) ) = x_1 POL( a__U12_2(x_1, x_2) ) = x_1 POL( isNatIListKind_1(x_1) ) = 0 POL( a__isNatIListKind_1(x_1) ) = 0 POL( U13_1(x_1) ) = x_1 POL( a__U13_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( U21_2(x_1, x_2) ) = x_1 POL( a__U21_2(x_1, x_2) ) = x_1 POL( U22_2(x_1, x_2) ) = x_1 POL( a__U22_2(x_1, x_2) ) = x_1 POL( isNatKind_1(x_1) ) = 0 POL( a__isNatKind_1(x_1) ) = 0 POL( U23_1(x_1) ) = x_1 POL( a__U23_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 0 POL( a__isNat_1(x_1) ) = 0 POL( U31_2(x_1, x_2) ) = x_1 POL( a__U31_2(x_1, x_2) ) = x_1 POL( U32_2(x_1, x_2) ) = x_1 POL( a__U32_2(x_1, x_2) ) = x_1 POL( U33_1(x_1) ) = x_1 POL( a__U33_1(x_1) ) = x_1 POL( U41_3(x_1, ..., x_3) ) = x_1 POL( a__U41_3(x_1, ..., x_3) ) = x_1 POL( U42_3(x_1, ..., x_3) ) = x_1 POL( a__U42_3(x_1, ..., x_3) ) = x_1 POL( U43_3(x_1, ..., x_3) ) = x_1 POL( a__U43_3(x_1, ..., x_3) ) = x_1 POL( U44_3(x_1, ..., x_3) ) = x_1 POL( a__U44_3(x_1, ..., x_3) ) = x_1 POL( U45_2(x_1, x_2) ) = x_1 POL( a__U45_2(x_1, x_2) ) = x_1 POL( U46_1(x_1) ) = x_1 POL( a__U46_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 0 POL( a__isNatIList_1(x_1) ) = 0 POL( U51_2(x_1, x_2) ) = x_1 POL( a__U51_2(x_1, x_2) ) = x_1 POL( U52_1(x_1) ) = x_1 POL( a__U52_1(x_1) ) = x_1 POL( U61_1(x_1) ) = x_1 POL( a__U61_1(x_1) ) = x_1 POL( U71_1(x_1) ) = x_1 POL( a__U71_1(x_1) ) = x_1 POL( U81_3(x_1, ..., x_3) ) = x_1 POL( a__U81_3(x_1, ..., x_3) ) = x_1 POL( U82_3(x_1, ..., x_3) ) = x_1 POL( a__U82_3(x_1, ..., x_3) ) = x_1 POL( U83_3(x_1, ..., x_3) ) = x_1 POL( a__U83_3(x_1, ..., x_3) ) = x_1 POL( U84_3(x_1, ..., x_3) ) = x_1 POL( a__U84_3(x_1, ..., x_3) ) = x_1 POL( U85_2(x_1, x_2) ) = x_1 POL( a__U85_2(x_1, x_2) ) = x_1 POL( U86_1(x_1) ) = x_1 POL( a__U86_1(x_1) ) = x_1 POL( U91_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 POL( a__U91_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 POL( U92_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 POL( a__U92_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 POL( U93_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 POL( a__U93_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 POL( U94_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( a__U94_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( length_1(x_1) ) = x_1 POL( a__length_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( 0 ) = 0 POL( tt ) = 0 POL( s_1(x_1) ) = x_1 POL( nil ) = 2 POL( MARK_1(x_1) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__isNatList(X) -> isNatList(X) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U12(X1, X2) -> U12(X1, X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U22(X1, X2) -> U22(X1, X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U32(X1, X2) -> U32(X1, X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U45(X1, X2) -> U45(X1, X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U85(X1, X2) -> U85(X1, X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__zeros -> cons(0, zeros) a__zeros -> zeros a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIList(X) -> isNatIList(X) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(U85(X1, X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) A__U94(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) A__U94(tt, L) -> MARK(L) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__U94(tt, L) -> MARK(L) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__LENGTH_1(x_1) ) = x_1 + 1 POL( A__U92_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( A__U91_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( a__U11_2(x_1, x_2) ) = 2x_1 POL( a__U12_2(x_1, x_2) ) = 2x_1 POL( a__U83_3(x_1, ..., x_3) ) = 2x_1 POL( a__U84_3(x_1, ..., x_3) ) = x_1 POL( A__U93_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( A__U94_2(x_1, x_2) ) = 2x_2 + 1 POL( a__U52_1(x_1) ) = x_1 POL( a__U61_1(x_1) ) = 2x_1 POL( a__U21_2(x_1, x_2) ) = 2x_1 POL( a__U22_2(x_1, x_2) ) = 2x_1 POL( a__U81_3(x_1, ..., x_3) ) = x_1 POL( a__U82_3(x_1, ..., x_3) ) = 2x_1 POL( a__isNatIListKind_1(x_1) ) = 0 POL( nil ) = 0 POL( tt ) = 0 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = 2x_2 POL( a__U51_2(x_1, x_2) ) = 2x_1 POL( a__isNatKind_1(x_1) ) = 0 POL( isNatIListKind_1(x_1) ) = 0 POL( a__U23_1(x_1) ) = x_1 POL( a__U85_2(x_1, x_2) ) = 2x_1 POL( a__isNat_1(x_1) ) = 0 POL( 0 ) = 0 POL( length_1(x_1) ) = x_1 + 2 POL( s_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 0 POL( a__U71_1(x_1) ) = x_1 POL( isNatKind_1(x_1) ) = 0 POL( mark_1(x_1) ) = x_1 POL( a__zeros ) = 0 POL( U11_2(x_1, x_2) ) = 2x_1 POL( U12_2(x_1, x_2) ) = 2x_1 POL( U13_1(x_1) ) = 2x_1 POL( a__U13_1(x_1) ) = 2x_1 POL( isNatList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( U21_2(x_1, x_2) ) = 2x_1 POL( U22_2(x_1, x_2) ) = 2x_1 POL( U23_1(x_1) ) = x_1 POL( U31_2(x_1, x_2) ) = x_1 POL( a__U31_2(x_1, x_2) ) = x_1 POL( U32_2(x_1, x_2) ) = x_1 POL( a__U32_2(x_1, x_2) ) = x_1 POL( U33_1(x_1) ) = x_1 POL( a__U33_1(x_1) ) = x_1 POL( U41_3(x_1, ..., x_3) ) = x_1 POL( a__U41_3(x_1, ..., x_3) ) = x_1 POL( U42_3(x_1, ..., x_3) ) = x_1 POL( a__U42_3(x_1, ..., x_3) ) = x_1 POL( U43_3(x_1, ..., x_3) ) = 2x_1 POL( a__U43_3(x_1, ..., x_3) ) = 2x_1 POL( U44_3(x_1, ..., x_3) ) = x_1 POL( a__U44_3(x_1, ..., x_3) ) = x_1 POL( U45_2(x_1, x_2) ) = 2x_1 POL( a__U45_2(x_1, x_2) ) = 2x_1 POL( U46_1(x_1) ) = 2x_1 POL( a__U46_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( a__isNatIList_1(x_1) ) = 0 POL( U51_2(x_1, x_2) ) = 2x_1 POL( U52_1(x_1) ) = x_1 POL( U61_1(x_1) ) = 2x_1 POL( U71_1(x_1) ) = x_1 POL( U81_3(x_1, ..., x_3) ) = x_1 POL( U82_3(x_1, ..., x_3) ) = 2x_1 POL( U83_3(x_1, ..., x_3) ) = 2x_1 POL( U84_3(x_1, ..., x_3) ) = x_1 POL( U85_2(x_1, x_2) ) = 2x_1 POL( U86_1(x_1) ) = 2x_1 POL( a__U86_1(x_1) ) = 2x_1 POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( a__U91_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U92_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( a__U92_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U93_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( a__U93_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U94_2(x_1, x_2) ) = 2x_2 + 2 POL( a__U94_2(x_1, x_2) ) = 2x_2 + 2 POL( a__length_1(x_1) ) = x_1 + 2 POL( MARK_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__isNatList(X) -> isNatList(X) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U12(X1, X2) -> U12(X1, X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U22(X1, X2) -> U22(X1, X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U32(X1, X2) -> U32(X1, X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U45(X1, X2) -> U45(X1, X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U85(X1, X2) -> U85(X1, X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__zeros -> cons(0, zeros) a__zeros -> zeros a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIList(X) -> isNatIList(X) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(U85(X1, X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) A__U94(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (22) Complex Obligation (AND) ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) A__U94(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__LENGTH_1(x_1) ) = 2x_1 + 1 POL( A__U93_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( A__U91_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 POL( a__U23_1(x_1) ) = 1 POL( a__U85_2(x_1, x_2) ) = x_2 POL( a__isNat_1(x_1) ) = x_1 POL( 0 ) = 1 POL( tt ) = 1 POL( length_1(x_1) ) = 2 POL( a__U11_2(x_1, x_2) ) = 1 POL( a__isNatIListKind_1(x_1) ) = 2 POL( s_1(x_1) ) = 1 POL( a__U21_2(x_1, x_2) ) = 1 POL( a__isNatKind_1(x_1) ) = 2x_1 + 2 POL( isNat_1(x_1) ) = x_1 POL( A__U94_2(x_1, x_2) ) = 2x_2 + 1 POL( a__U22_2(x_1, x_2) ) = 1 POL( a__U81_3(x_1, ..., x_3) ) = 2x_3 POL( a__U82_3(x_1, ..., x_3) ) = 2x_3 POL( A__U92_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( a__U51_2(x_1, x_2) ) = 2 POL( a__U71_1(x_1) ) = 1 POL( a__U12_2(x_1, x_2) ) = 1 POL( a__U83_3(x_1, ..., x_3) ) = 2x_3 POL( a__U84_3(x_1, ..., x_3) ) = x_3 POL( a__U61_1(x_1) ) = 2x_1 + 2 POL( isNatKind_1(x_1) ) = 2x_1 + 2 POL( mark_1(x_1) ) = x_1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( U11_2(x_1, x_2) ) = 1 POL( U12_2(x_1, x_2) ) = 1 POL( isNatIListKind_1(x_1) ) = 2 POL( U13_1(x_1) ) = 1 POL( a__U13_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = x_1 POL( a__isNatList_1(x_1) ) = x_1 POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = 1 POL( U23_1(x_1) ) = 1 POL( U31_2(x_1, x_2) ) = 2 POL( a__U31_2(x_1, x_2) ) = 2 POL( U32_2(x_1, x_2) ) = 1 POL( a__U32_2(x_1, x_2) ) = 1 POL( U33_1(x_1) ) = 1 POL( a__U33_1(x_1) ) = 1 POL( U41_3(x_1, ..., x_3) ) = 1 POL( a__U41_3(x_1, ..., x_3) ) = 1 POL( U42_3(x_1, ..., x_3) ) = 1 POL( a__U42_3(x_1, ..., x_3) ) = 1 POL( U43_3(x_1, ..., x_3) ) = 1 POL( a__U43_3(x_1, ..., x_3) ) = 1 POL( U44_3(x_1, ..., x_3) ) = 1 POL( a__U44_3(x_1, ..., x_3) ) = 1 POL( U45_2(x_1, x_2) ) = 1 POL( a__U45_2(x_1, x_2) ) = 1 POL( U46_1(x_1) ) = 1 POL( a__U46_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 2 POL( a__isNatIList_1(x_1) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U52_1(x_1) ) = 2 POL( a__U52_1(x_1) ) = 2 POL( U61_1(x_1) ) = 2x_1 + 2 POL( U71_1(x_1) ) = 1 POL( U81_3(x_1, ..., x_3) ) = 2x_3 POL( U82_3(x_1, ..., x_3) ) = 2x_3 POL( U83_3(x_1, ..., x_3) ) = 2x_3 POL( U84_3(x_1, ..., x_3) ) = x_3 POL( U85_2(x_1, x_2) ) = x_2 POL( U86_1(x_1) ) = x_1 POL( a__U86_1(x_1) ) = x_1 POL( U91_3(x_1, ..., x_3) ) = 2 POL( a__U91_3(x_1, ..., x_3) ) = 2 POL( U92_3(x_1, ..., x_3) ) = 2 POL( a__U92_3(x_1, ..., x_3) ) = 2 POL( U93_3(x_1, ..., x_3) ) = 1 POL( a__U93_3(x_1, ..., x_3) ) = 1 POL( U94_2(x_1, x_2) ) = 1 POL( a__U94_2(x_1, x_2) ) = 1 POL( a__length_1(x_1) ) = 2 POL( cons_2(x_1, x_2) ) = 2x_2 POL( nil ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__isNatList(X) -> isNatList(X) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U12(X1, X2) -> U12(X1, X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U22(X1, X2) -> U22(X1, X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U32(X1, X2) -> U32(X1, X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U45(X1, X2) -> U45(X1, X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U85(X1, X2) -> U85(X1, X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__zeros -> cons(0, zeros) a__zeros -> zeros a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIList(X) -> isNatIList(X) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) A__U94(tt, L) -> A__LENGTH(mark(L)) A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (27) TRUE ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X1, X2)) -> MARK(X1) MARK(U11(X1, X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(U85(X1, X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt) -> tt a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) a__U85(tt, V2) -> a__U86(a__isNatList(V2)) a__U86(tt) -> tt a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) a__U94(tt, L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(U23(X)) -> a__U23(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U85(X1, X2)) -> a__U85(mark(X1), X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2)) -> a__U94(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__isNatKind(X) -> isNatKind(X) a__U23(X) -> U23(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X) -> U71(X) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U85(X1, X2) -> U85(X1, X2) a__U86(X) -> U86(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2) -> U94(X1, X2) a__length(X) -> length(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X1, X2)) -> MARK(X1) MARK(U11(X1, X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(U85(X1, X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(s(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MARK(U12(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U11(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U13(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U21(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U22(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U23(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U31(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U32(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U33(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U41(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U42(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U43(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U44(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U45(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U46(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U51(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U52(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U61(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U71(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U81(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U82(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U83(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U84(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U85(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U86(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(s(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (32) YES