/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 1407 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 3 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 11 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) 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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 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(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) 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, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) 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)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) 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, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) 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(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) 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, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0) -> ok(0) 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)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) 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(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(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(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) 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_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U13(x_1, x_2, x_3)) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U14(x_1, x_2, x_3)) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U15(x_1, x_2)) -> U15(encArg(x_1), encArg(x_2)) encArg(cons_U16(x_1)) -> U16(encArg(x_1)) 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, x_3)) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U32(x_1, x_2, x_3)) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U33(x_1, x_2, x_3)) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U34(x_1, x_2, x_3)) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U35(x_1, x_2)) -> U35(encArg(x_1), encArg(x_2)) encArg(cons_U36(x_1)) -> U36(encArg(x_1)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(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, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1, x_2)) -> U72(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U82(x_1, x_2, x_3)) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U83(x_1, x_2, x_3)) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U84(x_1, x_2, x_3)) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_U92(x_1)) -> U92(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) 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_isNat(x_1) -> isNat(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U13(x_1, x_2, x_3) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U14(x_1, x_2, x_3) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U15(x_1, x_2) -> U15(encArg(x_1), encArg(x_2)) encode_U16(x_1) -> U16(encArg(x_1)) 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, x_3) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U32(x_1, x_2, x_3) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U33(x_1, x_2, x_3) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U34(x_1, x_2, x_3) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U35(x_1, x_2) -> U35(encArg(x_1), encArg(x_2)) encode_U36(x_1) -> U36(encArg(x_1)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_U42(x_1) -> U42(encArg(x_1)) encode_U51(x_1) -> U51(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, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1, x_2) -> U72(encArg(x_1), encArg(x_2)) encode_U81(x_1, x_2, x_3) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U82(x_1, x_2, x_3) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U83(x_1, x_2, x_3) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U84(x_1, x_2, x_3) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_U92(x_1) -> U92(encArg(x_1)) encode_0 -> 0 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) 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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 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(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) 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, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) 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)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) 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, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) 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(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) 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, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0) -> ok(0) 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)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) 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(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) 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_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U13(x_1, x_2, x_3)) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U14(x_1, x_2, x_3)) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U15(x_1, x_2)) -> U15(encArg(x_1), encArg(x_2)) encArg(cons_U16(x_1)) -> U16(encArg(x_1)) 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, x_3)) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U32(x_1, x_2, x_3)) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U33(x_1, x_2, x_3)) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U34(x_1, x_2, x_3)) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U35(x_1, x_2)) -> U35(encArg(x_1), encArg(x_2)) encArg(cons_U36(x_1)) -> U36(encArg(x_1)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(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, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1, x_2)) -> U72(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U82(x_1, x_2, x_3)) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U83(x_1, x_2, x_3)) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U84(x_1, x_2, x_3)) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_U92(x_1)) -> U92(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) 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_isNat(x_1) -> isNat(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U13(x_1, x_2, x_3) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U14(x_1, x_2, x_3) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U15(x_1, x_2) -> U15(encArg(x_1), encArg(x_2)) encode_U16(x_1) -> U16(encArg(x_1)) 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, x_3) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U32(x_1, x_2, x_3) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U33(x_1, x_2, x_3) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U34(x_1, x_2, x_3) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U35(x_1, x_2) -> U35(encArg(x_1), encArg(x_2)) encode_U36(x_1) -> U36(encArg(x_1)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_U42(x_1) -> U42(encArg(x_1)) encode_U51(x_1) -> U51(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, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1, x_2) -> U72(encArg(x_1), encArg(x_2)) encode_U81(x_1, x_2, x_3) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U82(x_1, x_2, x_3) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U83(x_1, x_2, x_3) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U84(x_1, x_2, x_3) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_U92(x_1) -> U92(encArg(x_1)) encode_0 -> 0 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) 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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 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(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) 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, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) 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)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) 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, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) 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(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) 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, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0) -> ok(0) 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)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) 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(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) 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_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U13(x_1, x_2, x_3)) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U14(x_1, x_2, x_3)) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U15(x_1, x_2)) -> U15(encArg(x_1), encArg(x_2)) encArg(cons_U16(x_1)) -> U16(encArg(x_1)) 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, x_3)) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U32(x_1, x_2, x_3)) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U33(x_1, x_2, x_3)) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U34(x_1, x_2, x_3)) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U35(x_1, x_2)) -> U35(encArg(x_1), encArg(x_2)) encArg(cons_U36(x_1)) -> U36(encArg(x_1)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(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, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1, x_2)) -> U72(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U82(x_1, x_2, x_3)) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U83(x_1, x_2, x_3)) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U84(x_1, x_2, x_3)) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_U92(x_1)) -> U92(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) 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_isNat(x_1) -> isNat(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U13(x_1, x_2, x_3) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U14(x_1, x_2, x_3) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U15(x_1, x_2) -> U15(encArg(x_1), encArg(x_2)) encode_U16(x_1) -> U16(encArg(x_1)) 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, x_3) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U32(x_1, x_2, x_3) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U33(x_1, x_2, x_3) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U34(x_1, x_2, x_3) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U35(x_1, x_2) -> U35(encArg(x_1), encArg(x_2)) encode_U36(x_1) -> U36(encArg(x_1)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_U42(x_1) -> U42(encArg(x_1)) encode_U51(x_1) -> U51(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, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1, x_2) -> U72(encArg(x_1), encArg(x_2)) encode_U81(x_1, x_2, x_3) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U82(x_1, x_2, x_3) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U83(x_1, x_2, x_3) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U84(x_1, x_2, x_3) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_U92(x_1) -> U92(encArg(x_1)) encode_0 -> 0 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) 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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 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(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) 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, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) 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)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) 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, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) 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(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) 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, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0) -> ok(0) 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)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) 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(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) 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_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U13(x_1, x_2, x_3)) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U14(x_1, x_2, x_3)) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U15(x_1, x_2)) -> U15(encArg(x_1), encArg(x_2)) encArg(cons_U16(x_1)) -> U16(encArg(x_1)) 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, x_3)) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U32(x_1, x_2, x_3)) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U33(x_1, x_2, x_3)) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U34(x_1, x_2, x_3)) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U35(x_1, x_2)) -> U35(encArg(x_1), encArg(x_2)) encArg(cons_U36(x_1)) -> U36(encArg(x_1)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(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, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1, x_2)) -> U72(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U82(x_1, x_2, x_3)) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U83(x_1, x_2, x_3)) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U84(x_1, x_2, x_3)) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_U92(x_1)) -> U92(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) 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_isNat(x_1) -> isNat(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U13(x_1, x_2, x_3) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U14(x_1, x_2, x_3) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U15(x_1, x_2) -> U15(encArg(x_1), encArg(x_2)) encode_U16(x_1) -> U16(encArg(x_1)) 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, x_3) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U32(x_1, x_2, x_3) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U33(x_1, x_2, x_3) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U34(x_1, x_2, x_3) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U35(x_1, x_2) -> U35(encArg(x_1), encArg(x_2)) encode_U36(x_1) -> U36(encArg(x_1)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_U42(x_1) -> U42(encArg(x_1)) encode_U51(x_1) -> U51(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, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1, x_2) -> U72(encArg(x_1), encArg(x_2)) encode_U81(x_1, x_2, x_3) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U82(x_1, x_2, x_3) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U83(x_1, x_2, x_3) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U84(x_1, x_2, x_3) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_U92(x_1) -> U92(encArg(x_1)) encode_0 -> 0 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 U23(mark(X)) ->^+ mark(U23(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / mark(X)]. 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) 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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 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(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) 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, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) 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)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) 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, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) 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(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) 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, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0) -> ok(0) 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)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) 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(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) 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_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U13(x_1, x_2, x_3)) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U14(x_1, x_2, x_3)) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U15(x_1, x_2)) -> U15(encArg(x_1), encArg(x_2)) encArg(cons_U16(x_1)) -> U16(encArg(x_1)) 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, x_3)) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U32(x_1, x_2, x_3)) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U33(x_1, x_2, x_3)) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U34(x_1, x_2, x_3)) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U35(x_1, x_2)) -> U35(encArg(x_1), encArg(x_2)) encArg(cons_U36(x_1)) -> U36(encArg(x_1)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(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, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1, x_2)) -> U72(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U82(x_1, x_2, x_3)) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U83(x_1, x_2, x_3)) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U84(x_1, x_2, x_3)) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_U92(x_1)) -> U92(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) 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_isNat(x_1) -> isNat(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U13(x_1, x_2, x_3) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U14(x_1, x_2, x_3) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U15(x_1, x_2) -> U15(encArg(x_1), encArg(x_2)) encode_U16(x_1) -> U16(encArg(x_1)) 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, x_3) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U32(x_1, x_2, x_3) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U33(x_1, x_2, x_3) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U34(x_1, x_2, x_3) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U35(x_1, x_2) -> U35(encArg(x_1), encArg(x_2)) encode_U36(x_1) -> U36(encArg(x_1)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_U42(x_1) -> U42(encArg(x_1)) encode_U51(x_1) -> U51(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, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1, x_2) -> U72(encArg(x_1), encArg(x_2)) encode_U81(x_1, x_2, x_3) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U82(x_1, x_2, x_3) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U83(x_1, x_2, x_3) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U84(x_1, x_2, x_3) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_U92(x_1) -> U92(encArg(x_1)) encode_0 -> 0 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) 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, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 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(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) 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, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) 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)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) 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, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) 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(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) 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, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0) -> ok(0) 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)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) 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(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) 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_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U13(x_1, x_2, x_3)) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U14(x_1, x_2, x_3)) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U15(x_1, x_2)) -> U15(encArg(x_1), encArg(x_2)) encArg(cons_U16(x_1)) -> U16(encArg(x_1)) 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, x_3)) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U32(x_1, x_2, x_3)) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U33(x_1, x_2, x_3)) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U34(x_1, x_2, x_3)) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U35(x_1, x_2)) -> U35(encArg(x_1), encArg(x_2)) encArg(cons_U36(x_1)) -> U36(encArg(x_1)) encArg(cons_U41(x_1, x_2)) -> U41(encArg(x_1), encArg(x_2)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(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, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1, x_2)) -> U72(encArg(x_1), encArg(x_2)) encArg(cons_U81(x_1, x_2, x_3)) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U82(x_1, x_2, x_3)) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U83(x_1, x_2, x_3)) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U84(x_1, x_2, x_3)) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_U91(x_1, x_2)) -> U91(encArg(x_1), encArg(x_2)) encArg(cons_U92(x_1)) -> U92(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatKind(x_1)) -> isNatKind(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_U101(x_1, x_2, x_3) -> U101(encArg(x_1), encArg(x_2), encArg(x_3)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) 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_isNat(x_1) -> isNat(encArg(x_1)) encode_U104(x_1, x_2, x_3) -> U104(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U13(x_1, x_2, x_3) -> U13(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U14(x_1, x_2, x_3) -> U14(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U15(x_1, x_2) -> U15(encArg(x_1), encArg(x_2)) encode_U16(x_1) -> U16(encArg(x_1)) 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, x_3) -> U31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U32(x_1, x_2, x_3) -> U32(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U33(x_1, x_2, x_3) -> U33(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U34(x_1, x_2, x_3) -> U34(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U35(x_1, x_2) -> U35(encArg(x_1), encArg(x_2)) encode_U36(x_1) -> U36(encArg(x_1)) encode_U41(x_1, x_2) -> U41(encArg(x_1), encArg(x_2)) encode_U42(x_1) -> U42(encArg(x_1)) encode_U51(x_1) -> U51(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, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1, x_2) -> U72(encArg(x_1), encArg(x_2)) encode_U81(x_1, x_2, x_3) -> U81(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U82(x_1, x_2, x_3) -> U82(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U83(x_1, x_2, x_3) -> U83(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U84(x_1, x_2, x_3) -> U84(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_U91(x_1, x_2) -> U91(encArg(x_1), encArg(x_2)) encode_U92(x_1) -> U92(encArg(x_1)) encode_0 -> 0 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