/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSToCSRProof [EQUIVALENT, 0 ms] (2) CSR (3) CSRRRRProof [EQUIVALENT, 168 ms] (4) CSR (5) CSRRRRProof [EQUIVALENT, 39 ms] (6) CSR (7) CSRRRRProof [EQUIVALENT, 24 ms] (8) CSR (9) CSDependencyPairsProof [EQUIVALENT, 151 ms] (10) QCSDP (11) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QCSDP (14) QCSUsableRulesProof [EQUIVALENT, 0 ms] (15) QCSDP (16) QCSDPMuMonotonicPoloProof [EQUIVALENT, 7 ms] (17) QCSDP (18) PIsEmptyProof [EQUIVALENT, 0 ms] (19) YES (20) QCSDP (21) QCSUsableRulesProof [EQUIVALENT, 0 ms] (22) QCSDP (23) QCSDPMuMonotonicPoloProof [EQUIVALENT, 75 ms] (24) QCSDP (25) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (26) QCSDP (27) QCSDPSubtermProof [EQUIVALENT, 0 ms] (28) QCSDP (29) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (30) TRUE (31) QCSDP (32) QCSDPSubtermProof [EQUIVALENT, 2 ms] (33) QCSDP (34) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (35) TRUE (36) QCSDP (37) QCSDPReductionPairProof [EQUIVALENT, 103 ms] (38) QCSDP (39) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (40) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2)) active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2)) active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2)) active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2)) active(U105(tt, V2)) -> mark(U106(isNatIList(V2))) active(U106(tt)) -> mark(tt) active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N)) active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N)) active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L)) active(U114(tt, L)) -> mark(s(length(L))) active(U12(tt, V1)) -> mark(U13(isNatList(V1))) active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL))) active(U122(tt)) -> mark(nil) active(U13(tt)) -> mark(tt) active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N)) active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N)) active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N)) active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N)) active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N)) active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) active(U32(tt, V)) -> mark(U33(isNatList(V))) active(U33(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) active(U46(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt)) -> mark(tt) active(U81(tt)) -> mark(tt) active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2)) active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2)) active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2)) active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2)) active(U95(tt, V2)) -> mark(U96(isNatList(V2))) active(U96(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1))) active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2)) active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N)) active(take(0, IL)) -> mark(U121(isNatIList(IL), IL)) active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(U105(X1, X2)) -> U105(active(X1), X2) active(U106(X)) -> U106(active(X)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X1, X2)) -> U12(active(X1), X2) active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3) active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3) active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3) active(U114(X1, X2)) -> U114(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U13(X)) -> U13(active(X)) active(U121(X1, X2)) -> U121(active(X1), X2) active(U122(X)) -> U122(active(X)) active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4) active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4) active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4) active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4) active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4) active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) active(U45(X1, X2)) -> U45(active(X1), X2) active(U46(X)) -> U46(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X)) -> U71(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3) active(U95(X1, X2)) -> U95(active(X1), X2) active(U96(X)) -> U96(active(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) U105(mark(X1), X2) -> mark(U105(X1, X2)) U106(mark(X)) -> mark(U106(X)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3)) U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3)) U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3)) U114(mark(X1), X2) -> mark(U114(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U13(mark(X)) -> mark(U13(X)) U121(mark(X1), X2) -> mark(U121(X1, X2)) U122(mark(X)) -> mark(U122(X)) U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4)) U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4)) U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4)) U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4)) U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4)) U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) U45(mark(X1), X2) -> mark(U45(X1, X2)) U46(mark(X)) -> mark(U46(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X)) -> mark(U71(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3)) U95(mark(X1), X2) -> mark(U95(X1, X2)) U96(mark(X)) -> mark(U96(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U106(X)) -> U106(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3)) proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3)) proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3)) proper(U114(X1, X2)) -> U114(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) proper(U122(X)) -> U122(proper(X)) proper(nil) -> ok(nil) proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) proper(U46(X)) -> U46(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X)) -> U71(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3)) proper(U95(X1, X2)) -> U95(proper(X1), proper(X2)) proper(U96(X)) -> U96(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U106(ok(X)) -> ok(U106(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3)) U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3)) U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3)) U114(ok(X1), ok(X2)) -> ok(U114(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U13(ok(X)) -> ok(U13(X)) isNatList(ok(X)) -> ok(isNatList(X)) U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) U122(ok(X)) -> ok(U122(X)) U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4)) U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4)) U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4)) U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4)) U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4)) U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) U46(ok(X)) -> ok(U46(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X)) -> ok(U71(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3)) U95(ok(X1), ok(X2)) -> ok(U95(X1, X2)) U96(ok(X)) -> ok(U96(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Q is empty. ---------------------------------------- (1) QTRSToCSRProof (EQUIVALENT) The following Q TRS is given: Q restricted rewrite system: The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2)) active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2)) active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2)) active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2)) active(U105(tt, V2)) -> mark(U106(isNatIList(V2))) active(U106(tt)) -> mark(tt) active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N)) active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N)) active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L)) active(U114(tt, L)) -> mark(s(length(L))) active(U12(tt, V1)) -> mark(U13(isNatList(V1))) active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL))) active(U122(tt)) -> mark(nil) active(U13(tt)) -> mark(tt) active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N)) active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N)) active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N)) active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N)) active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N)) active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) active(U32(tt, V)) -> mark(U33(isNatList(V))) active(U33(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) active(U46(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt)) -> mark(tt) active(U81(tt)) -> mark(tt) active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2)) active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2)) active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2)) active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2)) active(U95(tt, V2)) -> mark(U96(isNatList(V2))) active(U96(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) active(isNatIListKind(nil)) -> mark(tt) active(isNatIListKind(zeros)) -> mark(tt) active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1))) active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2)) active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N)) active(take(0, IL)) -> mark(U121(isNatIList(IL), IL)) active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(U105(X1, X2)) -> U105(active(X1), X2) active(U106(X)) -> U106(active(X)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X1, X2)) -> U12(active(X1), X2) active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3) active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3) active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3) active(U114(X1, X2)) -> U114(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U13(X)) -> U13(active(X)) active(U121(X1, X2)) -> U121(active(X1), X2) active(U122(X)) -> U122(active(X)) active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4) active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4) active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4) active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4) active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4) active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) active(U45(X1, X2)) -> U45(active(X1), X2) active(U46(X)) -> U46(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X)) -> U71(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3) active(U95(X1, X2)) -> U95(active(X1), X2) active(U96(X)) -> U96(active(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) U105(mark(X1), X2) -> mark(U105(X1, X2)) U106(mark(X)) -> mark(U106(X)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3)) U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3)) U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3)) U114(mark(X1), X2) -> mark(U114(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U13(mark(X)) -> mark(U13(X)) U121(mark(X1), X2) -> mark(U121(X1, X2)) U122(mark(X)) -> mark(U122(X)) U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4)) U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4)) U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4)) U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4)) U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4)) U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) U45(mark(X1), X2) -> mark(U45(X1, X2)) U46(mark(X)) -> mark(U46(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X)) -> mark(U71(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3)) U95(mark(X1), X2) -> mark(U95(X1, X2)) U96(mark(X)) -> mark(U96(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U106(X)) -> U106(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3)) proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3)) proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3)) proper(U114(X1, X2)) -> U114(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) proper(U122(X)) -> U122(proper(X)) proper(nil) -> ok(nil) proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) proper(U46(X)) -> U46(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X)) -> U71(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3)) proper(U95(X1, X2)) -> U95(proper(X1), proper(X2)) proper(U96(X)) -> U96(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U106(ok(X)) -> ok(U106(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3)) U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3)) U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3)) U114(ok(X1), ok(X2)) -> ok(U114(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U13(ok(X)) -> ok(U13(X)) isNatList(ok(X)) -> ok(isNatList(X)) U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) U122(ok(X)) -> ok(U122(X)) U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4)) U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4)) U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4)) U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4)) U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4)) U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) U46(ok(X)) -> ok(U46(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X)) -> ok(U71(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3)) U95(ok(X1), ok(X2)) -> ok(U95(X1, X2)) U96(ok(X)) -> ok(U96(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Q is empty. Special symbols used for the transformation (see [GM04]): top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1 The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set U121: {1} U122: {1} nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound). ---------------------------------------- (2) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U121(tt, IL) -> U122(isNatIListKind(IL)) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(L), L, N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set U121: {1} U122: {1} nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} ---------------------------------------- (3) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U121(tt, IL) -> U122(isNatIListKind(IL)) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(L), L, N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set U121: {1} U122: {1} nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U101(x_1, x_2, x_3)) = x_1 POL(U102(x_1, x_2, x_3)) = x_1 POL(U103(x_1, x_2, x_3)) = x_1 POL(U104(x_1, x_2, x_3)) = x_1 POL(U105(x_1, x_2)) = x_1 POL(U106(x_1)) = x_1 POL(U11(x_1, x_2)) = x_1 POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U114(x_1, x_2)) = x_1 + x_2 POL(U12(x_1, x_2)) = x_1 POL(U121(x_1, x_2)) = 1 + x_1 + x_2 POL(U122(x_1)) = x_1 POL(U13(x_1)) = x_1 POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U21(x_1, x_2)) = x_1 POL(U22(x_1, x_2)) = x_1 POL(U23(x_1)) = x_1 POL(U31(x_1, x_2)) = x_1 POL(U32(x_1, x_2)) = x_1 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = x_1 POL(U42(x_1, x_2, x_3)) = x_1 POL(U43(x_1, x_2, x_3)) = x_1 POL(U44(x_1, x_2, x_3)) = x_1 POL(U45(x_1, x_2)) = x_1 POL(U46(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = x_1 POL(U62(x_1)) = x_1 POL(U71(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(U91(x_1, x_2, x_3)) = x_1 POL(U92(x_1, x_2, x_3)) = x_1 POL(U93(x_1, x_2, x_3)) = x_1 POL(U94(x_1, x_2, x_3)) = x_1 POL(U95(x_1, x_2)) = x_1 POL(U96(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatIListKind(x_1)) = 0 POL(isNatKind(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 1 + x_1 + x_2 POL(tt) = 0 POL(zeros) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U121(tt, IL) -> U122(isNatIListKind(IL)) ---------------------------------------- (4) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(L), L, N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set U121: {1} U122: {1} nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} ---------------------------------------- (5) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U122(tt) -> nil U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(nil) -> 0 length(cons(N, L)) -> U111(isNatList(L), L, N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set U121: {1} U122: {1} nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U101(x_1, x_2, x_3)) = x_1 POL(U102(x_1, x_2, x_3)) = x_1 POL(U103(x_1, x_2, x_3)) = x_1 POL(U104(x_1, x_2, x_3)) = x_1 POL(U105(x_1, x_2)) = x_1 POL(U106(x_1)) = x_1 POL(U11(x_1, x_2)) = x_1 POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U114(x_1, x_2)) = x_1 + x_2 POL(U12(x_1, x_2)) = x_1 POL(U121(x_1, x_2)) = x_1 + x_2 POL(U122(x_1)) = 1 + x_1 POL(U13(x_1)) = x_1 POL(U131(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U132(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U133(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U134(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U135(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U136(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U21(x_1, x_2)) = x_1 POL(U22(x_1, x_2)) = x_1 POL(U23(x_1)) = x_1 POL(U31(x_1, x_2)) = x_1 POL(U32(x_1, x_2)) = x_1 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = x_1 POL(U42(x_1, x_2, x_3)) = x_1 POL(U43(x_1, x_2, x_3)) = x_1 POL(U44(x_1, x_2, x_3)) = x_1 POL(U45(x_1, x_2)) = x_1 POL(U46(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = x_1 POL(U62(x_1)) = x_1 POL(U71(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(U91(x_1, x_2, x_3)) = x_1 POL(U92(x_1, x_2, x_3)) = x_1 POL(U93(x_1, x_2, x_3)) = x_1 POL(U94(x_1, x_2, x_3)) = x_1 POL(U95(x_1, x_2)) = x_1 POL(U96(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 1 POL(isNatIList(x_1)) = 1 POL(isNatIListKind(x_1)) = 1 POL(isNatKind(x_1)) = 1 POL(isNatList(x_1)) = 1 POL(length(x_1)) = 1 + x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 1 + x_1 + x_2 POL(tt) = 1 POL(zeros) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U122(tt) -> nil length(nil) -> 0 ---------------------------------------- (6) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set U121: {1} nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} ---------------------------------------- (7) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(0, IL) -> U121(isNatIList(IL), IL) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set U121: {1} nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U101(x_1, x_2, x_3)) = x_1 POL(U102(x_1, x_2, x_3)) = x_1 POL(U103(x_1, x_2, x_3)) = x_1 POL(U104(x_1, x_2, x_3)) = x_1 POL(U105(x_1, x_2)) = x_1 POL(U106(x_1)) = x_1 POL(U11(x_1, x_2)) = x_1 POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U114(x_1, x_2)) = x_1 + x_2 POL(U12(x_1, x_2)) = x_1 POL(U121(x_1, x_2)) = x_1 + x_2 POL(U13(x_1)) = x_1 POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(U21(x_1, x_2)) = x_1 POL(U22(x_1, x_2)) = x_1 POL(U23(x_1)) = x_1 POL(U31(x_1, x_2)) = x_1 POL(U32(x_1, x_2)) = x_1 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = x_1 POL(U42(x_1, x_2, x_3)) = x_1 POL(U43(x_1, x_2, x_3)) = x_1 POL(U44(x_1, x_2, x_3)) = x_1 POL(U45(x_1, x_2)) = x_1 POL(U46(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = x_1 POL(U62(x_1)) = x_1 POL(U71(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(U91(x_1, x_2, x_3)) = x_1 POL(U92(x_1, x_2, x_3)) = x_1 POL(U93(x_1, x_2, x_3)) = x_1 POL(U94(x_1, x_2, x_3)) = x_1 POL(U95(x_1, x_2)) = x_1 POL(U96(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatIListKind(x_1)) = 0 POL(isNatKind(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = x_1 POL(nil) = 1 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 1 + x_1 + x_2 POL(tt) = 0 POL(zeros) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: take(0, IL) -> U121(isNatIList(IL), IL) ---------------------------------------- (8) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) The replacement map contains the following entries: zeros: empty set cons: {1} 0: empty set U101: {1} tt: empty set U102: {1} isNatKind: empty set U103: {1} isNatIListKind: empty set U104: {1} U105: {1} isNat: empty set U106: {1} isNatIList: empty set U11: {1} U12: {1} U111: {1} U112: {1} U113: {1} U114: {1} s: {1} length: {1} U13: {1} isNatList: empty set nil: empty set U131: {1} U132: {1} U133: {1} U134: {1} U135: {1} U136: {1} take: {1, 2} U21: {1} U22: {1} U23: {1} U31: {1} U32: {1} U33: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U61: {1} U62: {1} U71: {1} U81: {1} U91: {1} U92: {1} U93: {1} U94: {1} U95: {1} U96: {1} ---------------------------------------- (9) CSDependencyPairsProof (EQUIVALENT) Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem. ---------------------------------------- (10) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, U106'_1, LENGTH_1, U13'_1, U23'_1, U33'_1, U46'_1, U52'_1, U62'_1, U96'_1, U71'_1, U81'_1, TAKE_2} are replacing on all positions. For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U102'_3, U101'_3, U103'_3, U104'_3, U105'_2, U12'_2, U11'_2, U112'_3, U111'_3, U113'_3, U114'_2, U132'_4, U131'_4, U133'_4, U134'_4, U135'_4, U136'_4, U22'_2, U21'_2, U32'_2, U31'_2, U42'_3, U41'_3, U43'_3, U44'_3, U45'_2, U51'_2, U61'_2, U92'_3, U91'_3, U93'_3, U94'_3, U95'_2} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATKIND_1, ISNATILISTKIND_1, ISNAT_1, ISNATILIST_1, ISNATLIST_1, U_1} are not replacing on any position. The ordinary context-sensitive dependency pairs DP_o are: U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) U101'(tt, V1, V2) -> ISNATKIND(V1) U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) U102'(tt, V1, V2) -> ISNATILISTKIND(V2) U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) U103'(tt, V1, V2) -> ISNATILISTKIND(V2) U104'(tt, V1, V2) -> U105'(isNat(V1), V2) U104'(tt, V1, V2) -> ISNAT(V1) U105'(tt, V2) -> U106'(isNatIList(V2)) U105'(tt, V2) -> ISNATILIST(V2) U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) U11'(tt, V1) -> ISNATILISTKIND(V1) U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N) U111'(tt, L, N) -> ISNATILISTKIND(L) U112'(tt, L, N) -> U113'(isNat(N), L, N) U112'(tt, L, N) -> ISNAT(N) U113'(tt, L, N) -> U114'(isNatKind(N), L) U113'(tt, L, N) -> ISNATKIND(N) U114'(tt, L) -> LENGTH(L) U12'(tt, V1) -> U13'(isNatList(V1)) U12'(tt, V1) -> ISNATLIST(V1) U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) U131'(tt, IL, M, N) -> ISNATILISTKIND(IL) U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) U132'(tt, IL, M, N) -> ISNAT(M) U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) U133'(tt, IL, M, N) -> ISNATKIND(M) U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) U134'(tt, IL, M, N) -> ISNAT(N) U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) U135'(tt, IL, M, N) -> ISNATKIND(N) U21'(tt, V1) -> U22'(isNatKind(V1), V1) U21'(tt, V1) -> ISNATKIND(V1) U22'(tt, V1) -> U23'(isNat(V1)) U22'(tt, V1) -> ISNAT(V1) U31'(tt, V) -> U32'(isNatIListKind(V), V) U31'(tt, V) -> ISNATILISTKIND(V) U32'(tt, V) -> U33'(isNatList(V)) U32'(tt, V) -> ISNATLIST(V) U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) U41'(tt, V1, V2) -> ISNATKIND(V1) U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) U42'(tt, V1, V2) -> ISNATILISTKIND(V2) U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) U43'(tt, V1, V2) -> ISNATILISTKIND(V2) U44'(tt, V1, V2) -> U45'(isNat(V1), V2) U44'(tt, V1, V2) -> ISNAT(V1) U45'(tt, V2) -> U46'(isNatIList(V2)) U45'(tt, V2) -> ISNATILIST(V2) U51'(tt, V2) -> U52'(isNatIListKind(V2)) U51'(tt, V2) -> ISNATILISTKIND(V2) U61'(tt, V2) -> U62'(isNatIListKind(V2)) U61'(tt, V2) -> ISNATILISTKIND(V2) U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) U91'(tt, V1, V2) -> ISNATKIND(V1) U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) U92'(tt, V1, V2) -> ISNATILISTKIND(V2) U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) U93'(tt, V1, V2) -> ISNATILISTKIND(V2) U94'(tt, V1, V2) -> U95'(isNat(V1), V2) U94'(tt, V1, V2) -> ISNAT(V1) U95'(tt, V2) -> U96'(isNatList(V2)) U95'(tt, V2) -> ISNATLIST(V2) ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) ISNAT(length(V1)) -> ISNATILISTKIND(V1) ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) ISNAT(s(V1)) -> ISNATKIND(V1) ISNATILIST(V) -> U31'(isNatIListKind(V), V) ISNATILIST(V) -> ISNATILISTKIND(V) ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) ISNATILIST(cons(V1, V2)) -> ISNATKIND(V1) ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) ISNATKIND(length(V1)) -> U71'(isNatIListKind(V1)) ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) ISNATKIND(s(V1)) -> U81'(isNatKind(V1)) ISNATKIND(s(V1)) -> ISNATKIND(V1) ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) ISNATLIST(cons(V1, V2)) -> ISNATKIND(V1) ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) ISNATLIST(take(V1, V2)) -> ISNATKIND(V1) LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N) LENGTH(cons(N, L)) -> ISNATLIST(L) TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N) TAKE(s(M), cons(N, IL)) -> ISNATILIST(IL) The collapsing dependency pairs are DP_c: U114'(tt, L) -> L U136'(tt, IL, M, N) -> N The hidden terms of R are: zeros take(x0, x1) Every hiding context is built from: aprove.DPFramework.CSDPProblem.QCSDPProblem$1@a8f54b0 Hence, the new unhiding pairs DP_u are : U114'(tt, L) -> U(L) U136'(tt, IL, M, N) -> U(N) U(take(x_0, x_1)) -> U(x_0) U(take(x_0, x_1)) -> U(x_1) U(zeros) -> ZEROS U(take(x0, x1)) -> TAKE(x0, x1) The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) Q is empty. ---------------------------------------- (11) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 40 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions. For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U51'_2, U61'_2} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATILISTKIND_1, ISNATKIND_1} are not replacing on any position. The TRS P consists of the following rules: ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) U51'(tt, V2) -> ISNATILISTKIND(V2) ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) ISNATKIND(s(V1)) -> ISNATKIND(V1) ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) U61'(tt, V2) -> ISNATILISTKIND(V2) ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) Q is empty. ---------------------------------------- (14) QCSUsableRulesProof (EQUIVALENT) The following rules are not useable [DA_EMMES] and can be deleted: zeros -> cons(0, zeros) U101(tt, x0, x1) -> U102(isNatKind(x0), x0, x1) U102(tt, x0, x1) -> U103(isNatIListKind(x1), x0, x1) U103(tt, x0, x1) -> U104(isNatIListKind(x1), x0, x1) U104(tt, x0, x1) -> U105(isNat(x0), x1) U105(tt, x0) -> U106(isNatIList(x0)) U106(tt) -> tt U11(tt, x0) -> U12(isNatIListKind(x0), x0) U111(tt, x0, x1) -> U112(isNatIListKind(x0), x0, x1) U112(tt, x0, x1) -> U113(isNat(x1), x0, x1) U113(tt, x0, x1) -> U114(isNatKind(x1), x0) U114(tt, x0) -> s(length(x0)) U12(tt, x0) -> U13(isNatList(x0)) U13(tt) -> tt U131(tt, x0, x1, x2) -> U132(isNatIListKind(x0), x0, x1, x2) U132(tt, x0, x1, x2) -> U133(isNat(x1), x0, x1, x2) U133(tt, x0, x1, x2) -> U134(isNatKind(x1), x0, x1, x2) U134(tt, x0, x1, x2) -> U135(isNat(x2), x0, x1, x2) U135(tt, x0, x1, x2) -> U136(isNatKind(x2), x0, x1, x2) U136(tt, x0, x1, x2) -> cons(x2, take(x1, x0)) U21(tt, x0) -> U22(isNatKind(x0), x0) U22(tt, x0) -> U23(isNat(x0)) U23(tt) -> tt U31(tt, x0) -> U32(isNatIListKind(x0), x0) U32(tt, x0) -> U33(isNatList(x0)) U33(tt) -> tt U41(tt, x0, x1) -> U42(isNatKind(x0), x0, x1) U42(tt, x0, x1) -> U43(isNatIListKind(x1), x0, x1) U43(tt, x0, x1) -> U44(isNatIListKind(x1), x0, x1) U44(tt, x0, x1) -> U45(isNat(x0), x1) U45(tt, x0) -> U46(isNatIList(x0)) U46(tt) -> tt U91(tt, x0, x1) -> U92(isNatKind(x0), x0, x1) U92(tt, x0, x1) -> U93(isNatIListKind(x1), x0, x1) U93(tt, x0, x1) -> U94(isNatIListKind(x1), x0, x1) U94(tt, x0, x1) -> U95(isNat(x0), x1) U95(tt, x0) -> U96(isNatList(x0)) U96(tt) -> tt isNat(0) -> tt isNat(length(x0)) -> U11(isNatIListKind(x0), x0) isNat(s(x0)) -> U21(isNatKind(x0), x0) isNatIList(x0) -> U31(isNatIListKind(x0), x0) isNatIList(zeros) -> tt isNatIList(cons(x0, x1)) -> U41(isNatKind(x0), x0, x1) isNatList(nil) -> tt isNatList(cons(x0, x1)) -> U91(isNatKind(x0), x0, x1) isNatList(take(x0, x1)) -> U101(isNatKind(x0), x0, x1) length(cons(x0, x1)) -> U111(isNatList(x1), x1, x0) take(s(x0), cons(x1, x2)) -> U131(isNatIList(x2), x2, x0, x1) ---------------------------------------- (15) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {length_1, U71_1, s_1, U81_1, U52_1, take_2, U62_1} are replacing on all positions. For all symbols f in {cons_2, U51_2, U61_2, U51'_2, U61'_2} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, ISNATILISTKIND_1, ISNATKIND_1} are not replacing on any position. The TRS P consists of the following rules: ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) U51'(tt, V2) -> ISNATILISTKIND(V2) ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) ISNATKIND(s(V1)) -> ISNATKIND(V1) ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) U61'(tt, V2) -> ISNATILISTKIND(V2) ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) The TRS R consists of the following rules: isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatKind(s(V1)) -> U81(isNatKind(V1)) U81(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U52(tt) -> tt U71(tt) -> tt Q is empty. ---------------------------------------- (16) QCSDPMuMonotonicPoloProof (EQUIVALENT) By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted. Strictly oriented dependency pairs: ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) U51'(tt, V2) -> ISNATILISTKIND(V2) ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) ISNATKIND(s(V1)) -> ISNATKIND(V1) ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) U61'(tt, V2) -> ISNATILISTKIND(V2) ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) Strictly oriented rules of the TRS R: isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) U81(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U52(tt) -> tt U71(tt) -> tt Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 2 POL(ISNATILISTKIND(x_1)) = 1 + 2*x_1 POL(ISNATKIND(x_1)) = 2 + 2*x_1 POL(U51(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U51'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U52(x_1)) = 1 + 2*x_1 POL(U61(x_1, x_2)) = x_1 + 2*x_2 POL(U61'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U62(x_1)) = 1 + 2*x_1 POL(U71(x_1)) = 1 + 2*x_1 POL(U81(x_1)) = 1 + 2*x_1 POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(isNatIListKind(x_1)) = x_1 POL(isNatKind(x_1)) = 2*x_1 POL(length(x_1)) = 2 + 2*x_1 POL(nil) = 2 POL(s(x_1)) = 2 + 2*x_1 POL(take(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(tt) = 2 POL(zeros) = 2 ---------------------------------------- (17) Obligation: Q-restricted context-sensitive dependency pair problem: For all symbols f in {cons_2, U51_2} we have mu(f) = {1}. The symbols in {isNatIListKind_1, isNatKind_1} are not replacing on any position. The TRS P consists of the following rules: none The TRS R consists of the following rules: isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) Q is empty. ---------------------------------------- (18) PIsEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain. ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions. For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U103'_3, U102'_3, U104'_3, U105'_2, U31'_2, U32'_2, U91'_3, U92'_3, U93'_3, U94'_3, U95'_2, U101'_3, U11'_2, U12'_2, U21'_2, U22'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position. The TRS P consists of the following rules: U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) U104'(tt, V1, V2) -> U105'(isNat(V1), V2) U105'(tt, V2) -> ISNATILIST(V2) ISNATILIST(V) -> U31'(isNatIListKind(V), V) U31'(tt, V) -> U32'(isNatIListKind(V), V) U32'(tt, V) -> ISNATLIST(V) ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) U94'(tt, V1, V2) -> U95'(isNat(V1), V2) U95'(tt, V2) -> ISNATLIST(V2) ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) U94'(tt, V1, V2) -> ISNAT(V1) ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) U12'(tt, V1) -> ISNATLIST(V1) ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) U21'(tt, V1) -> U22'(isNatKind(V1), V1) U22'(tt, V1) -> ISNAT(V1) ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) U44'(tt, V1, V2) -> U45'(isNat(V1), V2) U45'(tt, V2) -> ISNATILIST(V2) U44'(tt, V1, V2) -> ISNAT(V1) U104'(tt, V1, V2) -> ISNAT(V1) The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) Q is empty. ---------------------------------------- (21) QCSUsableRulesProof (EQUIVALENT) The following rules are not useable [DA_EMMES] and can be deleted: zeros -> cons(0, zeros) U111(tt, x0, x1) -> U112(isNatIListKind(x0), x0, x1) U112(tt, x0, x1) -> U113(isNat(x1), x0, x1) U113(tt, x0, x1) -> U114(isNatKind(x1), x0) U114(tt, x0) -> s(length(x0)) U131(tt, x0, x1, x2) -> U132(isNatIListKind(x0), x0, x1, x2) U132(tt, x0, x1, x2) -> U133(isNat(x1), x0, x1, x2) U133(tt, x0, x1, x2) -> U134(isNatKind(x1), x0, x1, x2) U134(tt, x0, x1, x2) -> U135(isNat(x2), x0, x1, x2) U135(tt, x0, x1, x2) -> U136(isNatKind(x2), x0, x1, x2) U136(tt, x0, x1, x2) -> cons(x2, take(x1, x0)) length(cons(x0, x1)) -> U111(isNatList(x1), x1, x0) take(s(x0), cons(x1, x2)) -> U131(isNatIList(x2), x2, x0, x1) ---------------------------------------- (22) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U103'_3, U102'_3, U104'_3, U105'_2, U31'_2, U32'_2, U91'_3, U92'_3, U93'_3, U94'_3, U95'_2, U101'_3, U11'_2, U12'_2, U21'_2, U22'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2} we have mu(f) = {1}. The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position. The TRS P consists of the following rules: U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) U104'(tt, V1, V2) -> U105'(isNat(V1), V2) U105'(tt, V2) -> ISNATILIST(V2) ISNATILIST(V) -> U31'(isNatIListKind(V), V) U31'(tt, V) -> U32'(isNatIListKind(V), V) U32'(tt, V) -> ISNATLIST(V) ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) U94'(tt, V1, V2) -> U95'(isNat(V1), V2) U95'(tt, V2) -> ISNATLIST(V2) ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) U94'(tt, V1, V2) -> ISNAT(V1) ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) U12'(tt, V1) -> ISNATLIST(V1) ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) U21'(tt, V1) -> U22'(isNatKind(V1), V1) U22'(tt, V1) -> ISNAT(V1) ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) U44'(tt, V1, V2) -> U45'(isNat(V1), V2) U45'(tt, V2) -> ISNATILIST(V2) U44'(tt, V1, V2) -> ISNAT(V1) U104'(tt, V1, V2) -> ISNAT(V1) The TRS R consists of the following rules: isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(s(V1)) -> U81(isNatKind(V1)) U81(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) U11(tt, V1) -> U12(isNatIListKind(V1), V1) U12(tt, V1) -> U13(isNatList(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) isNat(s(V1)) -> U21(isNatKind(V1), V1) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U95(tt, V2) -> U96(isNatList(V2)) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) isNatIList(V) -> U31(isNatIListKind(V), V) U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U106(tt) -> tt U96(tt) -> tt U13(tt) -> tt Q is empty. ---------------------------------------- (23) QCSDPMuMonotonicPoloProof (EQUIVALENT) By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted. Strictly oriented dependency pairs: U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) U95'(tt, V2) -> ISNATLIST(V2) ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) U94'(tt, V1, V2) -> ISNAT(V1) ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) U12'(tt, V1) -> ISNATLIST(V1) ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) U45'(tt, V2) -> ISNATILIST(V2) U44'(tt, V1, V2) -> ISNAT(V1) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 0 POL(ISNAT(x_1)) = x_1 POL(ISNATILIST(x_1)) = 2*x_1 POL(ISNATLIST(x_1)) = 2*x_1 POL(U101(x_1, x_2, x_3)) = x_1 POL(U101'(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3 POL(U102(x_1, x_2, x_3)) = 2*x_1 POL(U102'(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3 POL(U103(x_1, x_2, x_3)) = 2*x_1 POL(U103'(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3 POL(U104(x_1, x_2, x_3)) = x_1 POL(U104'(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 POL(U105(x_1, x_2)) = 2*x_1 POL(U105'(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U106(x_1)) = x_1 POL(U11(x_1, x_2)) = x_1 POL(U11'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U12(x_1, x_2)) = 2*x_1 POL(U12'(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U13(x_1)) = 2*x_1 POL(U21(x_1, x_2)) = x_1 POL(U21'(x_1, x_2)) = x_1 + x_2 POL(U22(x_1, x_2)) = x_1 POL(U22'(x_1, x_2)) = x_1 + x_2 POL(U23(x_1)) = 2*x_1 POL(U31(x_1, x_2)) = 2*x_1 POL(U31'(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U32(x_1, x_2)) = x_1 POL(U32'(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U33(x_1)) = 2*x_1 POL(U41(x_1, x_2, x_3)) = x_1 POL(U41'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 POL(U42(x_1, x_2, x_3)) = 2*x_1 POL(U42'(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3 POL(U43(x_1, x_2, x_3)) = x_1 POL(U43'(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3 POL(U44(x_1, x_2, x_3)) = 2*x_1 POL(U44'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 POL(U45(x_1, x_2)) = x_1 POL(U45'(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U46(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = 2*x_1 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = 2*x_1 POL(U62(x_1)) = x_1 POL(U71(x_1)) = 2*x_1 POL(U81(x_1)) = 2*x_1 POL(U91(x_1, x_2, x_3)) = 2*x_1 POL(U91'(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 POL(U92(x_1, x_2, x_3)) = x_1 POL(U92'(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 POL(U93(x_1, x_2, x_3)) = x_1 POL(U93'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 POL(U94(x_1, x_2, x_3)) = x_1 POL(U94'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 POL(U95(x_1, x_2)) = 2*x_1 POL(U95'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U96(x_1)) = x_1 POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatIListKind(x_1)) = 0 POL(isNatKind(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 2 + 2*x_1 POL(nil) = 0 POL(s(x_1)) = 2*x_1 POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (24) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U104'_3, U103'_3, U105'_2, U31'_2, U32'_2, U92'_3, U91'_3, U94'_3, U93'_3, U95'_2, U12'_2, U11'_2, U21'_2, U22'_2, U42'_3, U41'_3, U44'_3, U43'_3, U45'_2} we have mu(f) = {1}. The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position. The TRS P consists of the following rules: U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) U104'(tt, V1, V2) -> U105'(isNat(V1), V2) U105'(tt, V2) -> ISNATILIST(V2) ISNATILIST(V) -> U31'(isNatIListKind(V), V) U31'(tt, V) -> U32'(isNatIListKind(V), V) U32'(tt, V) -> ISNATLIST(V) U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) U94'(tt, V1, V2) -> U95'(isNat(V1), V2) U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) U21'(tt, V1) -> U22'(isNatKind(V1), V1) U22'(tt, V1) -> ISNAT(V1) U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) U44'(tt, V1, V2) -> U45'(isNat(V1), V2) U104'(tt, V1, V2) -> ISNAT(V1) The TRS R consists of the following rules: isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(s(V1)) -> U81(isNatKind(V1)) U81(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) U11(tt, V1) -> U12(isNatIListKind(V1), V1) U12(tt, V1) -> U13(isNatList(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) isNat(s(V1)) -> U21(isNatKind(V1), V1) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U95(tt, V2) -> U96(isNatList(V2)) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) isNatIList(V) -> U31(isNatIListKind(V), V) U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U106(tt) -> tt U96(tt) -> tt U13(tt) -> tt Q is empty. ---------------------------------------- (25) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 14 less nodes. ---------------------------------------- (26) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U22'_2, U21'_2} we have mu(f) = {1}. The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNAT_1} are not replacing on any position. The TRS P consists of the following rules: U21'(tt, V1) -> U22'(isNatKind(V1), V1) U22'(tt, V1) -> ISNAT(V1) ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) The TRS R consists of the following rules: isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(s(V1)) -> U81(isNatKind(V1)) U81(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) U11(tt, V1) -> U12(isNatIListKind(V1), V1) U12(tt, V1) -> U13(isNatList(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) isNat(s(V1)) -> U21(isNatKind(V1), V1) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U95(tt, V2) -> U96(isNatList(V2)) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) isNatIList(V) -> U31(isNatIListKind(V), V) U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U106(tt) -> tt U96(tt) -> tt U13(tt) -> tt Q is empty. ---------------------------------------- (27) QCSDPSubtermProof (EQUIVALENT) We use the subterm processor [DA_EMMES]. The following pairs can be oriented strictly and are deleted. ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) The remaining pairs can at least be oriented weakly. U21'(tt, V1) -> U22'(isNatKind(V1), V1) U22'(tt, V1) -> ISNAT(V1) Used ordering: Combined order from the following AFS and order. U22'(x1, x2) = x2 U21'(x1, x2) = x2 ISNAT(x1) = x1 Subterm Order ---------------------------------------- (28) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U22'_2, U21'_2} we have mu(f) = {1}. The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNAT_1} are not replacing on any position. The TRS P consists of the following rules: U21'(tt, V1) -> U22'(isNatKind(V1), V1) U22'(tt, V1) -> ISNAT(V1) The TRS R consists of the following rules: isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(s(V1)) -> U81(isNatKind(V1)) U81(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) U11(tt, V1) -> U12(isNatIListKind(V1), V1) U12(tt, V1) -> U13(isNatList(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) isNat(s(V1)) -> U21(isNatKind(V1), V1) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U95(tt, V2) -> U96(isNatList(V2)) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) isNatIList(V) -> U31(isNatIListKind(V), V) U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U106(tt) -> tt U96(tt) -> tt U13(tt) -> tt Q is empty. ---------------------------------------- (29) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes. ---------------------------------------- (30) TRUE ---------------------------------------- (31) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, TAKE_2} are replacing on all positions. For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U133'_4, U132'_4, U134'_4, U135'_4, U136'_4, U131'_4} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, U_1} are not replacing on any position. The TRS P consists of the following rules: U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) U136'(tt, IL, M, N) -> U(N) U(take(x_0, x_1)) -> U(x_0) U(take(x_0, x_1)) -> U(x_1) U(take(x0, x1)) -> TAKE(x0, x1) TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N) U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) Q is empty. ---------------------------------------- (32) QCSDPSubtermProof (EQUIVALENT) We use the subterm processor [DA_EMMES]. The following pairs can be oriented strictly and are deleted. U(take(x_0, x_1)) -> U(x_0) U(take(x_0, x_1)) -> U(x_1) U(take(x0, x1)) -> TAKE(x0, x1) TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N) The remaining pairs can at least be oriented weakly. U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) U136'(tt, IL, M, N) -> U(N) U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) Used ordering: Combined order from the following AFS and order. U133'(x1, x2, x3, x4) = x4 U132'(x1, x2, x3, x4) = x4 U134'(x1, x2, x3, x4) = x4 U135'(x1, x2, x3, x4) = x4 U136'(x1, x2, x3, x4) = x4 U(x1) = x1 TAKE(x1, x2) = x2 U131'(x1, x2, x3, x4) = x4 Subterm Order ---------------------------------------- (33) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions. For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U133'_4, U132'_4, U134'_4, U135'_4, U136'_4, U131'_4} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, U_1} are not replacing on any position. The TRS P consists of the following rules: U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) U136'(tt, IL, M, N) -> U(N) U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) Q is empty. ---------------------------------------- (34) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes. ---------------------------------------- (35) TRUE ---------------------------------------- (36) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, LENGTH_1} are replacing on all positions. For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U113'_3, U112'_3, U114'_2, U111'_3} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1} are not replacing on any position. The TRS P consists of the following rules: U112'(tt, L, N) -> U113'(isNat(N), L, N) U113'(tt, L, N) -> U114'(isNatKind(N), L) U114'(tt, L) -> LENGTH(L) LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N) U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N) The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) Q is empty. ---------------------------------------- (37) QCSDPReductionPairProof (EQUIVALENT) Using the order Polynomial interpretation [POLO]: POL(0) = 2 POL(LENGTH(x_1)) = 2 + 2*x_1 POL(U101(x_1, x_2, x_3)) = x_2 POL(U102(x_1, x_2, x_3)) = x_2 POL(U103(x_1, x_2, x_3)) = x_2 POL(U104(x_1, x_2, x_3)) = x_2 POL(U105(x_1, x_2)) = x_1 POL(U106(x_1)) = 2 POL(U11(x_1, x_2)) = x_2 POL(U111(x_1, x_2, x_3)) = 2*x_2 POL(U111'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 POL(U112(x_1, x_2, x_3)) = 2*x_2 POL(U112'(x_1, x_2, x_3)) = 2 + 2*x_2 POL(U113(x_1, x_2, x_3)) = 2*x_2 POL(U113'(x_1, x_2, x_3)) = 2 + 2*x_2 POL(U114(x_1, x_2)) = 2*x_2 POL(U114'(x_1, x_2)) = 2 + 2*x_2 POL(U12(x_1, x_2)) = x_2 POL(U13(x_1)) = x_1 POL(U131(x_1, x_2, x_3, x_4)) = 2*x_3 POL(U132(x_1, x_2, x_3, x_4)) = 2*x_3 POL(U133(x_1, x_2, x_3, x_4)) = 2*x_3 POL(U134(x_1, x_2, x_3, x_4)) = 2*x_3 POL(U135(x_1, x_2, x_3, x_4)) = 2*x_3 POL(U136(x_1, x_2, x_3, x_4)) = 2*x_3 POL(U21(x_1, x_2)) = 2*x_2 POL(U22(x_1, x_2)) = 2*x_2 POL(U23(x_1)) = 2*x_1 POL(U31(x_1, x_2)) = 2*x_2 POL(U32(x_1, x_2)) = 2*x_2 POL(U33(x_1)) = 2*x_1 POL(U41(x_1, x_2, x_3)) = 2 + 2*x_3 POL(U42(x_1, x_2, x_3)) = 2 POL(U43(x_1, x_2, x_3)) = x_1 POL(U44(x_1, x_2, x_3)) = x_1 POL(U45(x_1, x_2)) = 2 POL(U46(x_1)) = 2 POL(U51(x_1, x_2)) = 2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = 2 POL(U62(x_1)) = 2 POL(U71(x_1)) = 2 POL(U81(x_1)) = 2 POL(U91(x_1, x_2, x_3)) = 2*x_3 POL(U92(x_1, x_2, x_3)) = x_3 POL(U93(x_1, x_2, x_3)) = x_3 POL(U94(x_1, x_2, x_3)) = x_3 POL(U95(x_1, x_2)) = x_2 POL(U96(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = 2 + 2*x_1 POL(isNatIListKind(x_1)) = 2 POL(isNatKind(x_1)) = 2 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = x_1 POL(nil) = 2 POL(s(x_1)) = 2*x_1 POL(take(x_1, x_2)) = x_1 POL(tt) = 2 POL(zeros) = 0 the following usable rules isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) length(cons(N, L)) -> U111(isNatList(L), L, N) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) U71(tt) -> tt isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) zeros -> cons(0, zeros) U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U81(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt could all be oriented weakly. Furthermore, the pairs LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N) U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N) could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES]. ---------------------------------------- (38) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, LENGTH_1} are replacing on all positions. For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U113'_3, U112'_3, U114'_2} we have mu(f) = {1}. The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1} are not replacing on any position. The TRS P consists of the following rules: U112'(tt, L, N) -> U113'(isNat(N), L, N) U113'(tt, L, N) -> U114'(isNatKind(N), L) U114'(tt, L) -> LENGTH(L) The TRS R consists of the following rules: zeros -> cons(0, zeros) U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNat(V1), V2) U105(tt, V2) -> U106(isNatIList(V2)) U106(tt) -> tt U11(tt, V1) -> U12(isNatIListKind(V1), V1) U111(tt, L, N) -> U112(isNatIListKind(L), L, N) U112(tt, L, N) -> U113(isNat(N), L, N) U113(tt, L, N) -> U114(isNatKind(N), L) U114(tt, L) -> s(length(L)) U12(tt, V1) -> U13(isNatList(V1)) U13(tt) -> tt U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) U136(tt, IL, M, N) -> cons(N, take(M, IL)) U21(tt, V1) -> U22(isNatKind(V1), V1) U22(tt, V1) -> U23(isNat(V1)) U23(tt) -> tt U31(tt, V) -> U32(isNatIListKind(V), V) U32(tt, V) -> U33(isNatList(V)) U33(tt) -> tt U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNat(V1), V2) U45(tt, V2) -> U46(isNatIList(V2)) U46(tt) -> tt U51(tt, V2) -> U52(isNatIListKind(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isNatIListKind(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) U94(tt, V1, V2) -> U95(isNat(V1), V2) U95(tt, V2) -> U96(isNatList(V2)) U96(tt) -> tt isNat(0) -> tt isNat(length(V1)) -> U11(isNatIListKind(V1), V1) isNat(s(V1)) -> U21(isNatKind(V1), V1) isNatIList(V) -> U31(isNatIListKind(V), V) isNatIList(zeros) -> tt isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) isNatIListKind(nil) -> tt isNatIListKind(zeros) -> tt isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) isNatKind(0) -> tt isNatKind(length(V1)) -> U71(isNatIListKind(V1)) isNatKind(s(V1)) -> U81(isNatKind(V1)) isNatList(nil) -> tt isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) length(cons(N, L)) -> U111(isNatList(L), L, N) take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) Q is empty. ---------------------------------------- (39) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes. ---------------------------------------- (40) TRUE