WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 17 ms] (2) CpxTRS (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 7 ms] (6) CdtProblem (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 10 ms] (12) CdtProblem (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 565 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 52 ms] (18) CdtProblem (19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (20) BOUNDS(1, 1) (21) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (22) TRS for Loop Detection (23) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (24) BEST (25) proven lower bound (26) LowerBoundPropagationProof [FINISHED, 0 ms] (27) BOUNDS(n^1, INF) (28) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 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: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 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)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 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)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: 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(tt) -> ok(tt) 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: FULL ---------------------------------------- (3) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: 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(tt) -> ok(tt) 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 ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) proper(tt) -> ok(tt) proper(0) -> ok(0) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) PROPER(tt) -> c72 PROPER(0) -> c73 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c77(TOP(active(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) PROPER(tt) -> c72 PROPER(0) -> c73 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c77(TOP(active(z0))) K tuples:none Defined Rule Symbols: U101_3, U102_3, U103_3, U104_3, plus_2, x_2, U11_3, U12_3, U13_3, U14_3, U15_2, U16_1, U21_2, U22_2, U23_1, U31_3, U32_3, U33_3, U34_3, U35_2, U36_1, U41_2, U42_1, U51_1, U61_2, U62_1, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, s_1, U91_2, U92_1, proper_1, isNatKind_1, isNat_1, top_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, PROPER_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72, c73, c74_1, c75_1, c76_2, c77_1 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: TOP(ok(z0)) -> c77(TOP(active(z0))) PROPER(tt) -> c72 PROPER(0) -> c73 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) proper(tt) -> ok(tt) proper(0) -> ok(0) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) K tuples:none Defined Rule Symbols: U101_3, U102_3, U103_3, U104_3, plus_2, x_2, U11_3, U12_3, U13_3, U14_3, U15_2, U16_1, U21_2, U22_2, U23_1, U31_3, U32_3, U33_3, U34_3, U35_2, U36_1, U41_2, U42_1, U51_1, U61_2, U62_1, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, s_1, U91_2, U92_1, proper_1, isNatKind_1, isNat_1, top_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_2 ---------------------------------------- (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) proper(tt) -> ok(tt) proper(0) -> ok(0) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples:none Defined Rule Symbols: U101_3, U102_3, U103_3, U104_3, plus_2, x_2, U11_3, U12_3, U13_3, U14_3, U15_2, U16_1, U21_2, U22_2, U23_1, U31_3, U32_3, U33_3, U34_3, U35_2, U36_1, U41_2, U42_1, U51_1, U61_2, U62_1, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, s_1, U91_2, U92_1, proper_1, isNatKind_1, isNat_1, top_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples:none Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) We considered the (Usable) Rules:none And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(ISNAT(x_1)) = 0 POL(ISNATKIND(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(TOP(x_1)) = 0 POL(U101'(x_1, x_2, x_3)) = 0 POL(U102'(x_1, x_2, x_3)) = x_1 POL(U103'(x_1, x_2, x_3)) = 0 POL(U104'(x_1, x_2, x_3)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U13'(x_1, x_2, x_3)) = 0 POL(U14'(x_1, x_2, x_3)) = 0 POL(U15'(x_1, x_2)) = 0 POL(U16'(x_1)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U23'(x_1)) = 0 POL(U31'(x_1, x_2, x_3)) = 0 POL(U32'(x_1, x_2, x_3)) = 0 POL(U33'(x_1, x_2, x_3)) = 0 POL(U34'(x_1, x_2, x_3)) = 0 POL(U35'(x_1, x_2)) = 0 POL(U36'(x_1)) = 0 POL(U41'(x_1, x_2)) = 0 POL(U42'(x_1)) = 0 POL(U51'(x_1)) = 0 POL(U61'(x_1, x_2)) = 0 POL(U62'(x_1)) = 0 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(U83'(x_1, x_2, x_3)) = 0 POL(U84'(x_1, x_2, x_3)) = 0 POL(U91'(x_1, x_2)) = 0 POL(U92'(x_1)) = 0 POL(X(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c65(x_1)) = x_1 POL(c66(x_1)) = x_1 POL(c67(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c69(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c70(x_1)) = x_1 POL(c71(x_1)) = x_1 POL(c74(x_1)) = x_1 POL(c75(x_1)) = x_1 POL(c76(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [2] + x_1 POL(ok(x_1)) = [2] + x_1 POL(proper(x_1)) = x_1 + [2]x_1^2 POL(tt) = [1] ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples: U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ISNAT(x_1)) = x_1 POL(ISNATKIND(x_1)) = x_1 POL(PLUS(x_1, x_2)) = x_1 + x_2 POL(S(x_1)) = x_1 POL(TOP(x_1)) = x_1 POL(U101'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U102'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U103'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U104'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U11'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U12'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U13'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U14'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U15'(x_1, x_2)) = x_1 + x_2 POL(U16'(x_1)) = x_1 POL(U21'(x_1, x_2)) = x_1 + x_2 POL(U22'(x_1, x_2)) = x_1 + x_2 POL(U23'(x_1)) = x_1 POL(U31'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U32'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U33'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U34'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U35'(x_1, x_2)) = x_1 + x_2 POL(U36'(x_1)) = x_1 POL(U41'(x_1, x_2)) = x_1 + x_2 POL(U42'(x_1)) = x_1 POL(U51'(x_1)) = x_1 POL(U61'(x_1, x_2)) = x_1 + x_2 POL(U62'(x_1)) = x_1 POL(U71'(x_1, x_2)) = x_1 + x_2 POL(U72'(x_1, x_2)) = x_1 + x_2 POL(U81'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U82'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U83'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U84'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U91'(x_1, x_2)) = x_1 + x_2 POL(U92'(x_1)) = x_1 POL(X(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c65(x_1)) = x_1 POL(c66(x_1)) = x_1 POL(c67(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c69(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c70(x_1)) = x_1 POL(c71(x_1)) = x_1 POL(c74(x_1)) = x_1 POL(c75(x_1)) = x_1 POL(c76(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = [1] + x_1 POL(tt) = [1] ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples: U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOP(mark(z0)) -> c76(TOP(proper(z0))) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ISNAT(x_1)) = 0 POL(ISNATKIND(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(TOP(x_1)) = x_1 POL(U101'(x_1, x_2, x_3)) = 0 POL(U102'(x_1, x_2, x_3)) = 0 POL(U103'(x_1, x_2, x_3)) = 0 POL(U104'(x_1, x_2, x_3)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U13'(x_1, x_2, x_3)) = 0 POL(U14'(x_1, x_2, x_3)) = 0 POL(U15'(x_1, x_2)) = 0 POL(U16'(x_1)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U23'(x_1)) = 0 POL(U31'(x_1, x_2, x_3)) = 0 POL(U32'(x_1, x_2, x_3)) = 0 POL(U33'(x_1, x_2, x_3)) = 0 POL(U34'(x_1, x_2, x_3)) = 0 POL(U35'(x_1, x_2)) = 0 POL(U36'(x_1)) = 0 POL(U41'(x_1, x_2)) = 0 POL(U42'(x_1)) = 0 POL(U51'(x_1)) = 0 POL(U61'(x_1, x_2)) = 0 POL(U62'(x_1)) = 0 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(U83'(x_1, x_2, x_3)) = 0 POL(U84'(x_1, x_2, x_3)) = 0 POL(U91'(x_1, x_2)) = 0 POL(U92'(x_1)) = 0 POL(X(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c65(x_1)) = x_1 POL(c66(x_1)) = x_1 POL(c67(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c69(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c70(x_1)) = x_1 POL(c71(x_1)) = x_1 POL(c74(x_1)) = x_1 POL(c75(x_1)) = x_1 POL(c76(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(ok(x_1)) = 0 POL(proper(x_1)) = 0 POL(tt) = 0 ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples:none K tuples: U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (19) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (20) BOUNDS(1, 1) ---------------------------------------- (21) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (22) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 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: FULL ---------------------------------------- (23) 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 [ ]. ---------------------------------------- (24) Complex Obligation (BEST) ---------------------------------------- (25) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 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: FULL ---------------------------------------- (26) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (27) BOUNDS(n^1, INF) ---------------------------------------- (28) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 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: FULL