WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 2214 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). 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)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U102(x_1, x_2, x_3)) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U103(x_1, x_2, x_3)) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U104(x_1, x_2, x_3)) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U105(x_1, x_2)) -> U105(encArg(x_1), encArg(x_2)) encArg(cons_U106(x_1)) -> U106(encArg(x_1)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(cons_U111(x_1, x_2, x_3)) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U112(x_1, x_2, x_3)) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U113(x_1, x_2, x_3)) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U114(x_1, x_2)) -> U114(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_U13(x_1)) -> U13(encArg(x_1)) encArg(cons_U121(x_1, x_2)) -> U121(encArg(x_1), encArg(x_2)) encArg(cons_U122(x_1)) -> U122(encArg(x_1)) encArg(cons_U131(x_1, x_2, x_3, x_4)) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U132(x_1, x_2, x_3, x_4)) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U133(x_1, x_2, x_3, x_4)) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U134(x_1, x_2, x_3, x_4)) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U135(x_1, x_2, x_3, x_4)) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U136(x_1, x_2, x_3, x_4)) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1, x_2)) -> U32(encArg(x_1), encArg(x_2)) encArg(cons_U33(x_1)) -> U33(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2, x_3)) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U43(x_1, x_2, x_3)) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U44(x_1, x_2, x_3)) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U45(x_1, x_2)) -> U45(encArg(x_1), encArg(x_2)) encArg(cons_U46(x_1)) -> U46(encArg(x_1)) encArg(cons_U51(x_1, x_2)) -> U51(encArg(x_1), encArg(x_2)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_U91(x_1, x_2, x_3)) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U92(x_1, x_2, x_3)) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U93(x_1, x_2, x_3)) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U94(x_1, x_2, x_3)) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U95(x_1, x_2)) -> U95(encArg(x_1), encArg(x_2)) encArg(cons_U96(x_1)) -> U96(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_U102(x_1, x_2, x_3) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_U103(x_1, x_2, x_3) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U105(x_1, x_2) -> U105(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U106(x_1) -> U106(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_U111(x_1, x_2, x_3) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U112(x_1, x_2, x_3) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U113(x_1, x_2, x_3) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U114(x_1, x_2) -> U114(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_U13(x_1) -> U13(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U121(x_1, x_2) -> U121(encArg(x_1), encArg(x_2)) encode_U122(x_1) -> U122(encArg(x_1)) encode_nil -> nil encode_U131(x_1, x_2, x_3, x_4) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U132(x_1, x_2, x_3, x_4) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U133(x_1, x_2, x_3, x_4) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U134(x_1, x_2, x_3, x_4) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U135(x_1, x_2, x_3, x_4) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U136(x_1, x_2, x_3, x_4) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1, x_2) -> U32(encArg(x_1), encArg(x_2)) encode_U33(x_1) -> U33(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2, x_3) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U43(x_1, x_2, x_3) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U44(x_1, x_2, x_3) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U45(x_1, x_2) -> U45(encArg(x_1), encArg(x_2)) encode_U46(x_1) -> U46(encArg(x_1)) encode_U51(x_1, x_2) -> U51(encArg(x_1), encArg(x_2)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U81(x_1) -> U81(encArg(x_1)) encode_U91(x_1, x_2, x_3) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U92(x_1, x_2, x_3) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U93(x_1, x_2, x_3) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U94(x_1, x_2, x_3) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U95(x_1, x_2) -> U95(encArg(x_1), encArg(x_2)) encode_U96(x_1) -> U96(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). 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)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U102(x_1, x_2, x_3)) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U103(x_1, x_2, x_3)) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U104(x_1, x_2, x_3)) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U105(x_1, x_2)) -> U105(encArg(x_1), encArg(x_2)) encArg(cons_U106(x_1)) -> U106(encArg(x_1)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(cons_U111(x_1, x_2, x_3)) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U112(x_1, x_2, x_3)) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U113(x_1, x_2, x_3)) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U114(x_1, x_2)) -> U114(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_U13(x_1)) -> U13(encArg(x_1)) encArg(cons_U121(x_1, x_2)) -> U121(encArg(x_1), encArg(x_2)) encArg(cons_U122(x_1)) -> U122(encArg(x_1)) encArg(cons_U131(x_1, x_2, x_3, x_4)) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U132(x_1, x_2, x_3, x_4)) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U133(x_1, x_2, x_3, x_4)) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U134(x_1, x_2, x_3, x_4)) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U135(x_1, x_2, x_3, x_4)) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U136(x_1, x_2, x_3, x_4)) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1, x_2)) -> U32(encArg(x_1), encArg(x_2)) encArg(cons_U33(x_1)) -> U33(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2, x_3)) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U43(x_1, x_2, x_3)) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U44(x_1, x_2, x_3)) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U45(x_1, x_2)) -> U45(encArg(x_1), encArg(x_2)) encArg(cons_U46(x_1)) -> U46(encArg(x_1)) encArg(cons_U51(x_1, x_2)) -> U51(encArg(x_1), encArg(x_2)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_U91(x_1, x_2, x_3)) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U92(x_1, x_2, x_3)) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U93(x_1, x_2, x_3)) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U94(x_1, x_2, x_3)) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U95(x_1, x_2)) -> U95(encArg(x_1), encArg(x_2)) encArg(cons_U96(x_1)) -> U96(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_U102(x_1, x_2, x_3) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_U103(x_1, x_2, x_3) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U105(x_1, x_2) -> U105(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U106(x_1) -> U106(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_U111(x_1, x_2, x_3) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U112(x_1, x_2, x_3) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U113(x_1, x_2, x_3) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U114(x_1, x_2) -> U114(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_U13(x_1) -> U13(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U121(x_1, x_2) -> U121(encArg(x_1), encArg(x_2)) encode_U122(x_1) -> U122(encArg(x_1)) encode_nil -> nil encode_U131(x_1, x_2, x_3, x_4) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U132(x_1, x_2, x_3, x_4) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U133(x_1, x_2, x_3, x_4) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U134(x_1, x_2, x_3, x_4) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U135(x_1, x_2, x_3, x_4) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U136(x_1, x_2, x_3, x_4) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1, x_2) -> U32(encArg(x_1), encArg(x_2)) encode_U33(x_1) -> U33(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2, x_3) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U43(x_1, x_2, x_3) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U44(x_1, x_2, x_3) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U45(x_1, x_2) -> U45(encArg(x_1), encArg(x_2)) encode_U46(x_1) -> U46(encArg(x_1)) encode_U51(x_1, x_2) -> U51(encArg(x_1), encArg(x_2)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U81(x_1) -> U81(encArg(x_1)) encode_U91(x_1, x_2, x_3) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U92(x_1, x_2, x_3) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U93(x_1, x_2, x_3) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U94(x_1, x_2, x_3) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U95(x_1, x_2) -> U95(encArg(x_1), encArg(x_2)) encode_U96(x_1) -> U96(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). 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)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U102(x_1, x_2, x_3)) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U103(x_1, x_2, x_3)) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U104(x_1, x_2, x_3)) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U105(x_1, x_2)) -> U105(encArg(x_1), encArg(x_2)) encArg(cons_U106(x_1)) -> U106(encArg(x_1)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(cons_U111(x_1, x_2, x_3)) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U112(x_1, x_2, x_3)) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U113(x_1, x_2, x_3)) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U114(x_1, x_2)) -> U114(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_U13(x_1)) -> U13(encArg(x_1)) encArg(cons_U121(x_1, x_2)) -> U121(encArg(x_1), encArg(x_2)) encArg(cons_U122(x_1)) -> U122(encArg(x_1)) encArg(cons_U131(x_1, x_2, x_3, x_4)) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U132(x_1, x_2, x_3, x_4)) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U133(x_1, x_2, x_3, x_4)) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U134(x_1, x_2, x_3, x_4)) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U135(x_1, x_2, x_3, x_4)) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U136(x_1, x_2, x_3, x_4)) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1, x_2)) -> U32(encArg(x_1), encArg(x_2)) encArg(cons_U33(x_1)) -> U33(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2, x_3)) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U43(x_1, x_2, x_3)) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U44(x_1, x_2, x_3)) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U45(x_1, x_2)) -> U45(encArg(x_1), encArg(x_2)) encArg(cons_U46(x_1)) -> U46(encArg(x_1)) encArg(cons_U51(x_1, x_2)) -> U51(encArg(x_1), encArg(x_2)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_U91(x_1, x_2, x_3)) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U92(x_1, x_2, x_3)) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U93(x_1, x_2, x_3)) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U94(x_1, x_2, x_3)) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U95(x_1, x_2)) -> U95(encArg(x_1), encArg(x_2)) encArg(cons_U96(x_1)) -> U96(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_U102(x_1, x_2, x_3) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_U103(x_1, x_2, x_3) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U105(x_1, x_2) -> U105(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U106(x_1) -> U106(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_U111(x_1, x_2, x_3) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U112(x_1, x_2, x_3) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U113(x_1, x_2, x_3) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U114(x_1, x_2) -> U114(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_U13(x_1) -> U13(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U121(x_1, x_2) -> U121(encArg(x_1), encArg(x_2)) encode_U122(x_1) -> U122(encArg(x_1)) encode_nil -> nil encode_U131(x_1, x_2, x_3, x_4) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U132(x_1, x_2, x_3, x_4) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U133(x_1, x_2, x_3, x_4) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U134(x_1, x_2, x_3, x_4) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U135(x_1, x_2, x_3, x_4) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U136(x_1, x_2, x_3, x_4) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1, x_2) -> U32(encArg(x_1), encArg(x_2)) encode_U33(x_1) -> U33(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2, x_3) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U43(x_1, x_2, x_3) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U44(x_1, x_2, x_3) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U45(x_1, x_2) -> U45(encArg(x_1), encArg(x_2)) encode_U46(x_1) -> U46(encArg(x_1)) encode_U51(x_1, x_2) -> U51(encArg(x_1), encArg(x_2)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U81(x_1) -> U81(encArg(x_1)) encode_U91(x_1, x_2, x_3) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U92(x_1, x_2, x_3) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U93(x_1, x_2, x_3) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U94(x_1, x_2, x_3) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U95(x_1, x_2) -> U95(encArg(x_1), encArg(x_2)) encode_U96(x_1) -> U96(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). 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)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U102(x_1, x_2, x_3)) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U103(x_1, x_2, x_3)) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U104(x_1, x_2, x_3)) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U105(x_1, x_2)) -> U105(encArg(x_1), encArg(x_2)) encArg(cons_U106(x_1)) -> U106(encArg(x_1)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(cons_U111(x_1, x_2, x_3)) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U112(x_1, x_2, x_3)) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U113(x_1, x_2, x_3)) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U114(x_1, x_2)) -> U114(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_U13(x_1)) -> U13(encArg(x_1)) encArg(cons_U121(x_1, x_2)) -> U121(encArg(x_1), encArg(x_2)) encArg(cons_U122(x_1)) -> U122(encArg(x_1)) encArg(cons_U131(x_1, x_2, x_3, x_4)) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U132(x_1, x_2, x_3, x_4)) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U133(x_1, x_2, x_3, x_4)) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U134(x_1, x_2, x_3, x_4)) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U135(x_1, x_2, x_3, x_4)) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U136(x_1, x_2, x_3, x_4)) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1, x_2)) -> U32(encArg(x_1), encArg(x_2)) encArg(cons_U33(x_1)) -> U33(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2, x_3)) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U43(x_1, x_2, x_3)) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U44(x_1, x_2, x_3)) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U45(x_1, x_2)) -> U45(encArg(x_1), encArg(x_2)) encArg(cons_U46(x_1)) -> U46(encArg(x_1)) encArg(cons_U51(x_1, x_2)) -> U51(encArg(x_1), encArg(x_2)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_U91(x_1, x_2, x_3)) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U92(x_1, x_2, x_3)) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U93(x_1, x_2, x_3)) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U94(x_1, x_2, x_3)) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U95(x_1, x_2)) -> U95(encArg(x_1), encArg(x_2)) encArg(cons_U96(x_1)) -> U96(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_U102(x_1, x_2, x_3) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_U103(x_1, x_2, x_3) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U105(x_1, x_2) -> U105(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U106(x_1) -> U106(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_U111(x_1, x_2, x_3) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U112(x_1, x_2, x_3) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U113(x_1, x_2, x_3) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U114(x_1, x_2) -> U114(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_U13(x_1) -> U13(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U121(x_1, x_2) -> U121(encArg(x_1), encArg(x_2)) encode_U122(x_1) -> U122(encArg(x_1)) encode_nil -> nil encode_U131(x_1, x_2, x_3, x_4) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U132(x_1, x_2, x_3, x_4) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U133(x_1, x_2, x_3, x_4) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U134(x_1, x_2, x_3, x_4) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U135(x_1, x_2, x_3, x_4) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U136(x_1, x_2, x_3, x_4) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1, x_2) -> U32(encArg(x_1), encArg(x_2)) encode_U33(x_1) -> U33(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2, x_3) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U43(x_1, x_2, x_3) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U44(x_1, x_2, x_3) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U45(x_1, x_2) -> U45(encArg(x_1), encArg(x_2)) encode_U46(x_1) -> U46(encArg(x_1)) encode_U51(x_1, x_2) -> U51(encArg(x_1), encArg(x_2)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U81(x_1) -> U81(encArg(x_1)) encode_U91(x_1, x_2, x_3) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U92(x_1, x_2, x_3) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U93(x_1, x_2, x_3) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U94(x_1, x_2, x_3) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U95(x_1, x_2) -> U95(encArg(x_1), encArg(x_2)) encode_U96(x_1) -> U96(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence U22(mark(X1), X2) ->^+ mark(U22(X1, X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / mark(X1)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). 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)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U102(x_1, x_2, x_3)) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U103(x_1, x_2, x_3)) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U104(x_1, x_2, x_3)) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U105(x_1, x_2)) -> U105(encArg(x_1), encArg(x_2)) encArg(cons_U106(x_1)) -> U106(encArg(x_1)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(cons_U111(x_1, x_2, x_3)) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U112(x_1, x_2, x_3)) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U113(x_1, x_2, x_3)) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U114(x_1, x_2)) -> U114(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_U13(x_1)) -> U13(encArg(x_1)) encArg(cons_U121(x_1, x_2)) -> U121(encArg(x_1), encArg(x_2)) encArg(cons_U122(x_1)) -> U122(encArg(x_1)) encArg(cons_U131(x_1, x_2, x_3, x_4)) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U132(x_1, x_2, x_3, x_4)) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U133(x_1, x_2, x_3, x_4)) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U134(x_1, x_2, x_3, x_4)) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U135(x_1, x_2, x_3, x_4)) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U136(x_1, x_2, x_3, x_4)) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1, x_2)) -> U32(encArg(x_1), encArg(x_2)) encArg(cons_U33(x_1)) -> U33(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2, x_3)) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U43(x_1, x_2, x_3)) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U44(x_1, x_2, x_3)) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U45(x_1, x_2)) -> U45(encArg(x_1), encArg(x_2)) encArg(cons_U46(x_1)) -> U46(encArg(x_1)) encArg(cons_U51(x_1, x_2)) -> U51(encArg(x_1), encArg(x_2)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_U91(x_1, x_2, x_3)) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U92(x_1, x_2, x_3)) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U93(x_1, x_2, x_3)) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U94(x_1, x_2, x_3)) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U95(x_1, x_2)) -> U95(encArg(x_1), encArg(x_2)) encArg(cons_U96(x_1)) -> U96(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_U102(x_1, x_2, x_3) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_U103(x_1, x_2, x_3) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U105(x_1, x_2) -> U105(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U106(x_1) -> U106(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_U111(x_1, x_2, x_3) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U112(x_1, x_2, x_3) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U113(x_1, x_2, x_3) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U114(x_1, x_2) -> U114(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_U13(x_1) -> U13(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U121(x_1, x_2) -> U121(encArg(x_1), encArg(x_2)) encode_U122(x_1) -> U122(encArg(x_1)) encode_nil -> nil encode_U131(x_1, x_2, x_3, x_4) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U132(x_1, x_2, x_3, x_4) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U133(x_1, x_2, x_3, x_4) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U134(x_1, x_2, x_3, x_4) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U135(x_1, x_2, x_3, x_4) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U136(x_1, x_2, x_3, x_4) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1, x_2) -> U32(encArg(x_1), encArg(x_2)) encode_U33(x_1) -> U33(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2, x_3) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U43(x_1, x_2, x_3) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U44(x_1, x_2, x_3) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U45(x_1, x_2) -> U45(encArg(x_1), encArg(x_2)) encode_U46(x_1) -> U46(encArg(x_1)) encode_U51(x_1, x_2) -> U51(encArg(x_1), encArg(x_2)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U81(x_1) -> U81(encArg(x_1)) encode_U91(x_1, x_2, x_3) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U92(x_1, x_2, x_3) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U93(x_1, x_2, x_3) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U94(x_1, x_2, x_3) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U95(x_1, x_2) -> U95(encArg(x_1), encArg(x_2)) encode_U96(x_1) -> U96(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). 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)) The (relative) TRS S consists of the following rules: encArg(zeros) -> zeros encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(tt) -> tt encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_U101(x_1, x_2, x_3)) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U102(x_1, x_2, x_3)) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U103(x_1, x_2, x_3)) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U104(x_1, x_2, x_3)) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U105(x_1, x_2)) -> U105(encArg(x_1), encArg(x_2)) encArg(cons_U106(x_1)) -> U106(encArg(x_1)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(cons_U111(x_1, x_2, x_3)) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U112(x_1, x_2, x_3)) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U113(x_1, x_2, x_3)) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U114(x_1, x_2)) -> U114(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_U13(x_1)) -> U13(encArg(x_1)) encArg(cons_U121(x_1, x_2)) -> U121(encArg(x_1), encArg(x_2)) encArg(cons_U122(x_1)) -> U122(encArg(x_1)) encArg(cons_U131(x_1, x_2, x_3, x_4)) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U132(x_1, x_2, x_3, x_4)) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U133(x_1, x_2, x_3, x_4)) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U134(x_1, x_2, x_3, x_4)) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U135(x_1, x_2, x_3, x_4)) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_U136(x_1, x_2, x_3, x_4)) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_U21(x_1, x_2)) -> U21(encArg(x_1), encArg(x_2)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1, x_2)) -> U32(encArg(x_1), encArg(x_2)) encArg(cons_U33(x_1)) -> U33(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2, x_3)) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U43(x_1, x_2, x_3)) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U44(x_1, x_2, x_3)) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U45(x_1, x_2)) -> U45(encArg(x_1), encArg(x_2)) encArg(cons_U46(x_1)) -> U46(encArg(x_1)) encArg(cons_U51(x_1, x_2)) -> U51(encArg(x_1), encArg(x_2)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_U91(x_1, x_2, x_3)) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U92(x_1, x_2, x_3)) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U93(x_1, x_2, x_3)) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U94(x_1, x_2, x_3)) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U95(x_1, x_2)) -> U95(encArg(x_1), encArg(x_2)) encArg(cons_U96(x_1)) -> U96(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNatIListKind(x_1)) -> isNatIListKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_zeros -> zeros encode_mark(x_1) -> mark(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_U102(x_1, x_2, x_3) -> U102(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatKind(x_1) -> isNatKind(encArg(x_1)) encode_U103(x_1, x_2, x_3) -> U103(encArg(x_1), encArg(x_2), encArg(x_3)) encode_isNatIListKind(x_1) -> isNatIListKind(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U105(x_1, x_2) -> U105(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_U106(x_1) -> U106(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_U111(x_1, x_2, x_3) -> U111(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U112(x_1, x_2, x_3) -> U112(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U113(x_1, x_2, x_3) -> U113(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U114(x_1, x_2) -> U114(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_U13(x_1) -> U13(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_U121(x_1, x_2) -> U121(encArg(x_1), encArg(x_2)) encode_U122(x_1) -> U122(encArg(x_1)) encode_nil -> nil encode_U131(x_1, x_2, x_3, x_4) -> U131(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U132(x_1, x_2, x_3, x_4) -> U132(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U133(x_1, x_2, x_3, x_4) -> U133(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U134(x_1, x_2, x_3, x_4) -> U134(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U135(x_1, x_2, x_3, x_4) -> U135(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U136(x_1, x_2, x_3, x_4) -> U136(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2) -> U21(encArg(x_1), encArg(x_2)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1, x_2) -> U32(encArg(x_1), encArg(x_2)) encode_U33(x_1) -> U33(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2, x_3) -> U42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U43(x_1, x_2, x_3) -> U43(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U44(x_1, x_2, x_3) -> U44(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U45(x_1, x_2) -> U45(encArg(x_1), encArg(x_2)) encode_U46(x_1) -> U46(encArg(x_1)) encode_U51(x_1, x_2) -> U51(encArg(x_1), encArg(x_2)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U81(x_1) -> U81(encArg(x_1)) encode_U91(x_1, x_2, x_3) -> U91(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U92(x_1, x_2, x_3) -> U92(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U93(x_1, x_2, x_3) -> U93(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U94(x_1, x_2, x_3) -> U94(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U95(x_1, x_2) -> U95(encArg(x_1), encArg(x_2)) encode_U96(x_1) -> U96(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST