50.15/13.78 WORST_CASE(Omega(n^1), O(n^1)) 50.15/13.80 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 50.15/13.80 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 50.15/13.80 50.15/13.80 50.15/13.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 50.15/13.80 50.15/13.80 (0) CpxTRS 50.15/13.80 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 5 ms] 50.15/13.80 (2) CpxTRS 50.15/13.80 (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 50.15/13.80 (4) CpxTRS 50.15/13.80 (5) CpxTrsToCdtProof [UPPER BOUND(ID), 6 ms] 50.15/13.80 (6) CdtProblem 50.15/13.80 (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 50.15/13.80 (8) CdtProblem 50.15/13.80 (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 50.15/13.80 (10) CdtProblem 50.15/13.80 (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] 50.15/13.80 (12) CdtProblem 50.15/13.80 (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 270 ms] 50.15/13.80 (14) CdtProblem 50.15/13.80 (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 69 ms] 50.15/13.80 (16) CdtProblem 50.15/13.80 (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] 50.15/13.80 (18) BOUNDS(1, 1) 50.15/13.80 (19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 50.15/13.80 (20) TRS for Loop Detection 50.15/13.80 (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 50.15/13.80 (22) BEST 50.15/13.80 (23) proven lower bound 50.15/13.80 (24) LowerBoundPropagationProof [FINISHED, 0 ms] 50.15/13.80 (25) BOUNDS(n^1, INF) 50.15/13.80 (26) TRS for Loop Detection 50.15/13.80 50.15/13.80 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (0) 50.15/13.80 Obligation: 50.15/13.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 50.15/13.80 50.15/13.80 50.15/13.80 The TRS R consists of the following rules: 50.15/13.80 50.15/13.80 active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) 50.15/13.80 active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) 50.15/13.80 active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) 50.15/13.80 active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) 50.15/13.80 active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) 50.15/13.80 active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) 50.15/13.80 active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) 50.15/13.80 active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) 50.15/13.80 active(U15(tt, V2)) -> mark(U16(isNat(V2))) 50.15/13.80 active(U16(tt)) -> mark(tt) 50.15/13.80 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 50.15/13.80 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 50.15/13.80 active(U23(tt)) -> mark(tt) 50.15/13.80 active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) 50.15/13.80 active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) 50.15/13.80 active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) 50.15/13.80 active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) 50.15/13.80 active(U35(tt, V2)) -> mark(U36(isNat(V2))) 50.15/13.80 active(U36(tt)) -> mark(tt) 50.15/13.80 active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) 50.15/13.80 active(U42(tt)) -> mark(tt) 50.15/13.80 active(U51(tt)) -> mark(tt) 50.15/13.80 active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) 50.15/13.80 active(U62(tt)) -> mark(tt) 50.15/13.80 active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) 50.15/13.80 active(U72(tt, N)) -> mark(N) 50.15/13.80 active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) 50.15/13.80 active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) 50.15/13.80 active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) 50.15/13.80 active(U84(tt, M, N)) -> mark(s(plus(N, M))) 50.15/13.80 active(U91(tt, N)) -> mark(U92(isNatKind(N))) 50.15/13.80 active(U92(tt)) -> mark(0) 50.15/13.80 active(isNat(0)) -> mark(tt) 50.15/13.80 active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) 50.15/13.80 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 50.15/13.80 active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) 50.15/13.80 active(isNatKind(0)) -> mark(tt) 50.15/13.80 active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) 50.15/13.80 active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) 50.15/13.80 active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 50.15/13.80 active(plus(N, 0)) -> mark(U71(isNat(N), N)) 50.15/13.80 active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) 50.15/13.80 active(x(N, 0)) -> mark(U91(isNat(N), N)) 50.15/13.80 active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 50.15/13.80 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 50.15/13.80 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 50.15/13.80 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 50.15/13.80 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 50.15/13.80 active(plus(X1, X2)) -> plus(active(X1), X2) 50.15/13.80 active(plus(X1, X2)) -> plus(X1, active(X2)) 50.15/13.80 active(x(X1, X2)) -> x(active(X1), X2) 50.15/13.80 active(x(X1, X2)) -> x(X1, active(X2)) 50.15/13.80 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 50.15/13.80 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 50.15/13.80 active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) 50.15/13.80 active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) 50.15/13.80 active(U15(X1, X2)) -> U15(active(X1), X2) 50.15/13.80 active(U16(X)) -> U16(active(X)) 50.15/13.80 active(U21(X1, X2)) -> U21(active(X1), X2) 50.15/13.80 active(U22(X1, X2)) -> U22(active(X1), X2) 50.15/13.80 active(U23(X)) -> U23(active(X)) 50.15/13.80 active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) 50.15/13.80 active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) 50.15/13.80 active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) 50.15/13.80 active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) 50.15/13.80 active(U35(X1, X2)) -> U35(active(X1), X2) 50.15/13.80 active(U36(X)) -> U36(active(X)) 50.15/13.80 active(U41(X1, X2)) -> U41(active(X1), X2) 50.15/13.80 active(U42(X)) -> U42(active(X)) 50.15/13.80 active(U51(X)) -> U51(active(X)) 50.15/13.80 active(U61(X1, X2)) -> U61(active(X1), X2) 50.15/13.80 active(U62(X)) -> U62(active(X)) 50.15/13.80 active(U71(X1, X2)) -> U71(active(X1), X2) 50.15/13.80 active(U72(X1, X2)) -> U72(active(X1), X2) 50.15/13.80 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 50.15/13.80 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 50.15/13.80 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 50.15/13.80 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 50.15/13.80 active(s(X)) -> s(active(X)) 50.15/13.80 active(U91(X1, X2)) -> U91(active(X1), X2) 50.15/13.80 active(U92(X)) -> U92(active(X)) 50.15/13.80 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 50.15/13.80 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 50.15/13.80 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 50.15/13.80 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 50.15/13.80 plus(mark(X1), X2) -> mark(plus(X1, X2)) 50.15/13.80 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 50.15/13.80 x(mark(X1), X2) -> mark(x(X1, X2)) 50.15/13.80 x(X1, mark(X2)) -> mark(x(X1, X2)) 50.15/13.80 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 50.15/13.80 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 50.15/13.80 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 50.15/13.80 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 50.15/13.80 U15(mark(X1), X2) -> mark(U15(X1, X2)) 50.15/13.80 U16(mark(X)) -> mark(U16(X)) 50.15/13.80 U21(mark(X1), X2) -> mark(U21(X1, X2)) 50.15/13.80 U22(mark(X1), X2) -> mark(U22(X1, X2)) 50.15/13.80 U23(mark(X)) -> mark(U23(X)) 50.15/13.80 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 50.15/13.80 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 50.15/13.80 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 50.15/13.80 U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) 50.15/13.80 U35(mark(X1), X2) -> mark(U35(X1, X2)) 50.15/13.80 U36(mark(X)) -> mark(U36(X)) 50.15/13.80 U41(mark(X1), X2) -> mark(U41(X1, X2)) 50.15/13.80 U42(mark(X)) -> mark(U42(X)) 50.15/13.80 U51(mark(X)) -> mark(U51(X)) 50.15/13.80 U61(mark(X1), X2) -> mark(U61(X1, X2)) 50.15/13.80 U62(mark(X)) -> mark(U62(X)) 50.15/13.80 U71(mark(X1), X2) -> mark(U71(X1, X2)) 50.15/13.80 U72(mark(X1), X2) -> mark(U72(X1, X2)) 50.15/13.80 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 50.15/13.80 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 50.15/13.80 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 50.15/13.80 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 50.15/13.80 s(mark(X)) -> mark(s(X)) 50.15/13.80 U91(mark(X1), X2) -> mark(U91(X1, X2)) 50.15/13.80 U92(mark(X)) -> mark(U92(X)) 50.15/13.80 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(tt) -> ok(tt) 50.15/13.80 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(isNatKind(X)) -> isNatKind(proper(X)) 50.15/13.80 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(isNat(X)) -> isNat(proper(X)) 50.15/13.80 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 50.15/13.80 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 50.15/13.80 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) 50.15/13.80 proper(U16(X)) -> U16(proper(X)) 50.15/13.80 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 50.15/13.80 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 50.15/13.80 proper(U23(X)) -> U23(proper(X)) 50.15/13.80 proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) 50.15/13.80 proper(U36(X)) -> U36(proper(X)) 50.15/13.80 proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) 50.15/13.80 proper(U42(X)) -> U42(proper(X)) 50.15/13.80 proper(U51(X)) -> U51(proper(X)) 50.15/13.80 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 50.15/13.80 proper(U62(X)) -> U62(proper(X)) 50.15/13.80 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 50.15/13.80 proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) 50.15/13.80 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(s(X)) -> s(proper(X)) 50.15/13.80 proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) 50.15/13.80 proper(U92(X)) -> U92(proper(X)) 50.15/13.80 proper(0) -> ok(0) 50.15/13.80 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 50.15/13.80 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 50.15/13.80 isNatKind(ok(X)) -> ok(isNatKind(X)) 50.15/13.80 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 50.15/13.80 isNat(ok(X)) -> ok(isNat(X)) 50.15/13.80 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 50.15/13.80 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 50.15/13.80 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 50.15/13.80 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 50.15/13.80 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 50.15/13.80 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 50.15/13.80 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 50.15/13.80 U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) 50.15/13.80 U16(ok(X)) -> ok(U16(X)) 50.15/13.80 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 50.15/13.80 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 50.15/13.80 U23(ok(X)) -> ok(U23(X)) 50.15/13.80 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 50.15/13.80 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 50.15/13.80 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 50.15/13.80 U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) 50.15/13.80 U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) 50.15/13.80 U36(ok(X)) -> ok(U36(X)) 50.15/13.80 U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) 50.15/13.80 U42(ok(X)) -> ok(U42(X)) 50.15/13.80 U51(ok(X)) -> ok(U51(X)) 50.15/13.80 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 50.15/13.80 U62(ok(X)) -> ok(U62(X)) 50.15/13.80 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 50.15/13.80 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 50.15/13.80 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 50.15/13.80 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 50.15/13.80 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 50.15/13.80 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 50.15/13.80 s(ok(X)) -> ok(s(X)) 50.15/13.80 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 50.15/13.80 U92(ok(X)) -> ok(U92(X)) 50.15/13.80 top(mark(X)) -> top(proper(X)) 50.15/13.80 top(ok(X)) -> top(active(X)) 50.15/13.80 50.15/13.80 S is empty. 50.15/13.80 Rewrite Strategy: FULL 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 50.15/13.80 The following defined symbols can occur below the 0th argument of top: proper, active 50.15/13.80 The following defined symbols can occur below the 0th argument of proper: proper, active 50.15/13.80 The following defined symbols can occur below the 0th argument of active: proper, active 50.15/13.80 50.15/13.80 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 50.15/13.80 active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) 50.15/13.80 active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) 50.15/13.80 active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) 50.15/13.80 active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) 50.15/13.80 active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) 50.15/13.80 active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) 50.15/13.80 active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) 50.15/13.80 active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) 50.15/13.80 active(U15(tt, V2)) -> mark(U16(isNat(V2))) 50.15/13.80 active(U16(tt)) -> mark(tt) 50.15/13.80 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 50.15/13.80 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 50.15/13.80 active(U23(tt)) -> mark(tt) 50.15/13.80 active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) 50.15/13.80 active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) 50.15/13.80 active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) 50.15/13.80 active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) 50.15/13.80 active(U35(tt, V2)) -> mark(U36(isNat(V2))) 50.15/13.80 active(U36(tt)) -> mark(tt) 50.15/13.80 active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) 50.15/13.80 active(U42(tt)) -> mark(tt) 50.15/13.80 active(U51(tt)) -> mark(tt) 50.15/13.80 active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) 50.15/13.80 active(U62(tt)) -> mark(tt) 50.15/13.80 active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) 50.15/13.80 active(U72(tt, N)) -> mark(N) 50.15/13.80 active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) 50.15/13.80 active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) 50.15/13.80 active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) 50.15/13.80 active(U84(tt, M, N)) -> mark(s(plus(N, M))) 50.15/13.80 active(U91(tt, N)) -> mark(U92(isNatKind(N))) 50.15/13.80 active(U92(tt)) -> mark(0) 50.15/13.80 active(isNat(0)) -> mark(tt) 50.15/13.80 active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) 50.15/13.80 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 50.15/13.80 active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) 50.15/13.80 active(isNatKind(0)) -> mark(tt) 50.15/13.80 active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) 50.15/13.80 active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) 50.15/13.80 active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 50.15/13.80 active(plus(N, 0)) -> mark(U71(isNat(N), N)) 50.15/13.80 active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) 50.15/13.80 active(x(N, 0)) -> mark(U91(isNat(N), N)) 50.15/13.80 active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 50.15/13.80 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 50.15/13.80 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 50.15/13.80 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 50.15/13.80 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 50.15/13.80 active(plus(X1, X2)) -> plus(active(X1), X2) 50.15/13.80 active(plus(X1, X2)) -> plus(X1, active(X2)) 50.15/13.80 active(x(X1, X2)) -> x(active(X1), X2) 50.15/13.80 active(x(X1, X2)) -> x(X1, active(X2)) 50.15/13.80 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 50.15/13.80 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 50.15/13.80 active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) 50.15/13.80 active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) 50.15/13.80 active(U15(X1, X2)) -> U15(active(X1), X2) 50.15/13.80 active(U16(X)) -> U16(active(X)) 50.15/13.80 active(U21(X1, X2)) -> U21(active(X1), X2) 50.15/13.80 active(U22(X1, X2)) -> U22(active(X1), X2) 50.15/13.80 active(U23(X)) -> U23(active(X)) 50.15/13.80 active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) 50.15/13.80 active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) 50.15/13.80 active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) 50.15/13.80 active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) 50.15/13.80 active(U35(X1, X2)) -> U35(active(X1), X2) 50.15/13.80 active(U36(X)) -> U36(active(X)) 50.15/13.80 active(U41(X1, X2)) -> U41(active(X1), X2) 50.15/13.80 active(U42(X)) -> U42(active(X)) 50.15/13.80 active(U51(X)) -> U51(active(X)) 50.15/13.80 active(U61(X1, X2)) -> U61(active(X1), X2) 50.15/13.80 active(U62(X)) -> U62(active(X)) 50.15/13.80 active(U71(X1, X2)) -> U71(active(X1), X2) 50.15/13.80 active(U72(X1, X2)) -> U72(active(X1), X2) 50.15/13.80 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 50.15/13.80 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 50.15/13.80 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 50.15/13.80 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 50.15/13.80 active(s(X)) -> s(active(X)) 50.15/13.80 active(U91(X1, X2)) -> U91(active(X1), X2) 50.15/13.80 active(U92(X)) -> U92(active(X)) 50.15/13.80 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(isNatKind(X)) -> isNatKind(proper(X)) 50.15/13.80 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(isNat(X)) -> isNat(proper(X)) 50.15/13.80 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 50.15/13.80 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 50.15/13.80 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) 50.15/13.80 proper(U16(X)) -> U16(proper(X)) 50.15/13.80 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 50.15/13.80 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 50.15/13.80 proper(U23(X)) -> U23(proper(X)) 50.15/13.80 proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) 50.15/13.80 proper(U36(X)) -> U36(proper(X)) 50.15/13.80 proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) 50.15/13.80 proper(U42(X)) -> U42(proper(X)) 50.15/13.80 proper(U51(X)) -> U51(proper(X)) 50.15/13.80 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 50.15/13.80 proper(U62(X)) -> U62(proper(X)) 50.15/13.80 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 50.15/13.80 proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) 50.15/13.80 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 50.15/13.80 proper(s(X)) -> s(proper(X)) 50.15/13.80 proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) 50.15/13.80 proper(U92(X)) -> U92(proper(X)) 50.15/13.80 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (2) 50.15/13.80 Obligation: 50.15/13.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 50.15/13.80 50.15/13.80 50.15/13.80 The TRS R consists of the following rules: 50.15/13.80 50.15/13.80 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 50.15/13.80 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 50.15/13.80 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 50.15/13.80 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 50.15/13.80 plus(mark(X1), X2) -> mark(plus(X1, X2)) 50.15/13.80 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 50.15/13.80 x(mark(X1), X2) -> mark(x(X1, X2)) 50.15/13.80 x(X1, mark(X2)) -> mark(x(X1, X2)) 50.15/13.80 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 50.15/13.80 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 50.15/13.80 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 50.15/13.80 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 50.15/13.80 U15(mark(X1), X2) -> mark(U15(X1, X2)) 50.15/13.80 U16(mark(X)) -> mark(U16(X)) 50.15/13.80 U21(mark(X1), X2) -> mark(U21(X1, X2)) 50.15/13.80 U22(mark(X1), X2) -> mark(U22(X1, X2)) 50.15/13.80 U23(mark(X)) -> mark(U23(X)) 50.15/13.80 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 50.15/13.80 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 50.15/13.80 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 50.15/13.80 U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) 50.15/13.80 U35(mark(X1), X2) -> mark(U35(X1, X2)) 50.15/13.80 U36(mark(X)) -> mark(U36(X)) 50.15/13.80 U41(mark(X1), X2) -> mark(U41(X1, X2)) 50.15/13.80 U42(mark(X)) -> mark(U42(X)) 50.15/13.80 U51(mark(X)) -> mark(U51(X)) 50.15/13.80 U61(mark(X1), X2) -> mark(U61(X1, X2)) 50.15/13.80 U62(mark(X)) -> mark(U62(X)) 50.15/13.80 U71(mark(X1), X2) -> mark(U71(X1, X2)) 50.15/13.80 U72(mark(X1), X2) -> mark(U72(X1, X2)) 50.15/13.80 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 50.15/13.80 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 50.15/13.80 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 50.15/13.80 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 50.15/13.80 s(mark(X)) -> mark(s(X)) 50.15/13.80 U91(mark(X1), X2) -> mark(U91(X1, X2)) 50.15/13.80 U92(mark(X)) -> mark(U92(X)) 50.15/13.80 proper(tt) -> ok(tt) 50.15/13.80 proper(0) -> ok(0) 50.15/13.80 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 50.15/13.80 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 50.15/13.80 isNatKind(ok(X)) -> ok(isNatKind(X)) 50.15/13.80 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 50.15/13.80 isNat(ok(X)) -> ok(isNat(X)) 50.15/13.80 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 50.15/13.80 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 50.15/13.80 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 50.15/13.80 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 50.15/13.80 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 50.15/13.80 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 50.15/13.80 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 50.15/13.80 U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) 50.15/13.80 U16(ok(X)) -> ok(U16(X)) 50.15/13.80 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 50.15/13.80 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 50.15/13.80 U23(ok(X)) -> ok(U23(X)) 50.15/13.80 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 50.15/13.80 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 50.15/13.80 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 50.15/13.80 U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) 50.15/13.80 U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) 50.15/13.80 U36(ok(X)) -> ok(U36(X)) 50.15/13.80 U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) 50.15/13.80 U42(ok(X)) -> ok(U42(X)) 50.15/13.80 U51(ok(X)) -> ok(U51(X)) 50.15/13.80 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 50.15/13.80 U62(ok(X)) -> ok(U62(X)) 50.15/13.80 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 50.15/13.80 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 50.15/13.80 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 50.15/13.80 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 50.15/13.80 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 50.15/13.80 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 50.15/13.80 s(ok(X)) -> ok(s(X)) 50.15/13.80 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 50.15/13.80 U92(ok(X)) -> ok(U92(X)) 50.15/13.80 top(mark(X)) -> top(proper(X)) 50.15/13.80 top(ok(X)) -> top(active(X)) 50.15/13.80 50.15/13.80 S is empty. 50.15/13.80 Rewrite Strategy: FULL 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (3) RcToIrcProof (BOTH BOUNDS(ID, ID)) 50.15/13.80 Converted rc-obligation to irc-obligation. 50.15/13.80 50.15/13.80 As the TRS is a non-duplicating overlay system, we have rc = irc. 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (4) 50.15/13.80 Obligation: 50.15/13.80 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 50.15/13.80 50.15/13.80 50.15/13.80 The TRS R consists of the following rules: 50.15/13.80 50.15/13.80 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 50.15/13.80 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 50.15/13.80 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 50.15/13.80 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 50.15/13.80 plus(mark(X1), X2) -> mark(plus(X1, X2)) 50.15/13.80 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 50.15/13.80 x(mark(X1), X2) -> mark(x(X1, X2)) 50.15/13.80 x(X1, mark(X2)) -> mark(x(X1, X2)) 50.15/13.80 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 50.15/13.80 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 50.15/13.80 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 50.15/13.80 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 50.15/13.80 U15(mark(X1), X2) -> mark(U15(X1, X2)) 50.15/13.80 U16(mark(X)) -> mark(U16(X)) 50.15/13.80 U21(mark(X1), X2) -> mark(U21(X1, X2)) 50.15/13.80 U22(mark(X1), X2) -> mark(U22(X1, X2)) 50.15/13.80 U23(mark(X)) -> mark(U23(X)) 50.15/13.80 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 50.15/13.80 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 50.15/13.80 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 50.15/13.80 U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) 50.15/13.80 U35(mark(X1), X2) -> mark(U35(X1, X2)) 50.15/13.80 U36(mark(X)) -> mark(U36(X)) 50.15/13.80 U41(mark(X1), X2) -> mark(U41(X1, X2)) 50.15/13.80 U42(mark(X)) -> mark(U42(X)) 50.15/13.80 U51(mark(X)) -> mark(U51(X)) 50.15/13.80 U61(mark(X1), X2) -> mark(U61(X1, X2)) 50.15/13.80 U62(mark(X)) -> mark(U62(X)) 50.15/13.80 U71(mark(X1), X2) -> mark(U71(X1, X2)) 50.15/13.80 U72(mark(X1), X2) -> mark(U72(X1, X2)) 50.15/13.80 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 50.15/13.80 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 50.15/13.80 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 50.15/13.80 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 50.15/13.80 s(mark(X)) -> mark(s(X)) 50.15/13.80 U91(mark(X1), X2) -> mark(U91(X1, X2)) 50.15/13.80 U92(mark(X)) -> mark(U92(X)) 50.15/13.80 proper(tt) -> ok(tt) 50.15/13.80 proper(0) -> ok(0) 50.15/13.80 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 50.15/13.80 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 50.15/13.80 isNatKind(ok(X)) -> ok(isNatKind(X)) 50.15/13.80 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 50.15/13.80 isNat(ok(X)) -> ok(isNat(X)) 50.15/13.80 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 50.15/13.80 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 50.15/13.80 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 50.15/13.80 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 50.15/13.80 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 50.15/13.80 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 50.15/13.80 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 50.15/13.80 U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) 50.15/13.80 U16(ok(X)) -> ok(U16(X)) 50.15/13.80 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 50.15/13.80 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 50.15/13.80 U23(ok(X)) -> ok(U23(X)) 50.15/13.80 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 50.15/13.80 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 50.15/13.80 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 50.15/13.80 U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) 50.15/13.80 U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) 50.15/13.80 U36(ok(X)) -> ok(U36(X)) 50.15/13.80 U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) 50.15/13.80 U42(ok(X)) -> ok(U42(X)) 50.15/13.80 U51(ok(X)) -> ok(U51(X)) 50.15/13.80 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 50.15/13.80 U62(ok(X)) -> ok(U62(X)) 50.15/13.80 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 50.15/13.80 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 50.15/13.80 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 50.15/13.80 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 50.15/13.80 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 50.15/13.80 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 50.15/13.80 s(ok(X)) -> ok(s(X)) 50.15/13.80 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 50.15/13.80 U92(ok(X)) -> ok(U92(X)) 50.15/13.80 top(mark(X)) -> top(proper(X)) 50.15/13.80 top(ok(X)) -> top(active(X)) 50.15/13.80 50.15/13.80 S is empty. 50.15/13.80 Rewrite Strategy: INNERMOST 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (5) CpxTrsToCdtProof (UPPER BOUND(ID)) 50.15/13.80 Converted Cpx (relative) TRS to CDT 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (6) 50.15/13.80 Obligation: 50.15/13.80 Complexity Dependency Tuples Problem 50.15/13.80 50.15/13.80 Rules: 50.15/13.80 U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) 50.15/13.80 U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) 50.15/13.80 U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) 50.15/13.80 U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) 50.15/13.80 U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) 50.15/13.80 U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) 50.15/13.80 U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) 50.15/13.80 U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) 50.15/13.80 plus(mark(z0), z1) -> mark(plus(z0, z1)) 50.15/13.80 plus(z0, mark(z1)) -> mark(plus(z0, z1)) 50.15/13.80 plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) 50.15/13.80 x(mark(z0), z1) -> mark(x(z0, z1)) 50.15/13.80 x(z0, mark(z1)) -> mark(x(z0, z1)) 50.15/13.80 x(ok(z0), ok(z1)) -> ok(x(z0, z1)) 50.15/13.80 U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) 50.15/13.80 U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) 50.15/13.80 U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) 50.15/13.80 U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) 50.15/13.80 U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) 50.15/13.80 U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) 50.15/13.80 U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) 50.15/13.80 U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) 50.15/13.80 U15(mark(z0), z1) -> mark(U15(z0, z1)) 50.15/13.80 U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) 50.15/13.80 U16(mark(z0)) -> mark(U16(z0)) 50.15/13.80 U16(ok(z0)) -> ok(U16(z0)) 50.15/13.80 U21(mark(z0), z1) -> mark(U21(z0, z1)) 50.15/13.80 U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) 50.15/13.80 U22(mark(z0), z1) -> mark(U22(z0, z1)) 50.15/13.80 U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) 50.15/13.80 U23(mark(z0)) -> mark(U23(z0)) 50.15/13.80 U23(ok(z0)) -> ok(U23(z0)) 50.15/13.80 U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) 50.15/13.80 U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) 50.15/13.80 U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) 50.15/13.80 U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) 50.15/13.80 U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) 50.15/13.80 U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) 50.15/13.80 U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) 50.15/13.80 U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) 50.15/13.80 U35(mark(z0), z1) -> mark(U35(z0, z1)) 50.15/13.80 U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) 50.15/13.80 U36(mark(z0)) -> mark(U36(z0)) 50.15/13.80 U36(ok(z0)) -> ok(U36(z0)) 50.15/13.80 U41(mark(z0), z1) -> mark(U41(z0, z1)) 50.15/13.80 U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) 50.15/13.80 U42(mark(z0)) -> mark(U42(z0)) 50.15/13.80 U42(ok(z0)) -> ok(U42(z0)) 50.15/13.80 U51(mark(z0)) -> mark(U51(z0)) 50.15/13.80 U51(ok(z0)) -> ok(U51(z0)) 50.15/13.80 U61(mark(z0), z1) -> mark(U61(z0, z1)) 50.15/13.80 U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) 50.15/13.80 U62(mark(z0)) -> mark(U62(z0)) 50.15/13.80 U62(ok(z0)) -> ok(U62(z0)) 50.15/13.80 U71(mark(z0), z1) -> mark(U71(z0, z1)) 50.15/13.80 U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) 50.15/13.80 U72(mark(z0), z1) -> mark(U72(z0, z1)) 50.15/13.80 U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) 50.15/13.80 U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) 50.15/13.80 U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) 50.15/13.80 U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) 50.15/13.80 U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) 50.15/13.80 U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) 50.15/13.80 U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) 50.15/13.80 U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) 50.15/13.80 U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) 50.15/13.80 s(mark(z0)) -> mark(s(z0)) 50.15/13.80 s(ok(z0)) -> ok(s(z0)) 50.15/13.80 U91(mark(z0), z1) -> mark(U91(z0, z1)) 50.15/13.80 U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) 50.15/13.80 U92(mark(z0)) -> mark(U92(z0)) 50.15/13.80 U92(ok(z0)) -> ok(U92(z0)) 50.15/13.80 proper(tt) -> ok(tt) 50.15/13.80 proper(0) -> ok(0) 50.15/13.80 isNatKind(ok(z0)) -> ok(isNatKind(z0)) 50.15/13.80 isNat(ok(z0)) -> ok(isNat(z0)) 50.15/13.80 top(mark(z0)) -> top(proper(z0)) 50.15/13.80 top(ok(z0)) -> top(active(z0)) 50.15/13.80 Tuples: 50.15/13.80 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.80 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.80 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.80 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.80 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.80 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.80 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.80 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.80 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.80 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.80 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.80 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.80 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.80 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.80 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.80 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.80 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.80 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.80 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.80 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.80 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.80 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.80 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.80 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.80 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.80 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.80 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.80 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.80 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.80 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.80 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.80 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.80 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.80 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.80 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.80 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.80 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.80 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.80 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.80 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.80 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.80 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.80 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.80 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.80 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.80 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.80 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.80 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.80 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.80 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.80 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.80 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.80 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.80 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.80 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.80 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.80 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.80 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.80 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.80 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.80 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.80 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.80 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.80 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.80 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.80 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.80 S(mark(z0)) -> c66(S(z0)) 50.15/13.80 S(ok(z0)) -> c67(S(z0)) 50.15/13.80 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.80 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.80 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.80 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.80 PROPER(tt) -> c72 50.15/13.80 PROPER(0) -> c73 50.15/13.80 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.80 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.80 TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) 50.15/13.80 TOP(ok(z0)) -> c77(TOP(active(z0))) 50.15/13.80 S tuples: 50.15/13.80 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.80 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.80 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.80 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.80 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.80 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.80 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.80 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.80 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.80 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.80 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.80 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.80 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.80 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.80 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.80 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.80 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.80 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.80 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.80 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.80 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.80 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.80 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.80 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.80 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.80 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.80 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.80 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.80 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.80 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.80 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.80 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.80 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.80 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.80 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.80 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.80 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.80 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.80 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.80 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.80 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.80 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.80 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.80 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.80 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.80 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.80 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.80 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.80 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.80 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.80 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.80 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.80 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.80 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.80 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.80 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.80 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.80 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.80 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.80 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.80 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.80 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.80 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.80 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.80 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.80 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.80 S(mark(z0)) -> c66(S(z0)) 50.15/13.80 S(ok(z0)) -> c67(S(z0)) 50.15/13.80 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.80 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.80 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.80 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.80 PROPER(tt) -> c72 50.15/13.80 PROPER(0) -> c73 50.15/13.80 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.80 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.80 TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) 50.15/13.80 TOP(ok(z0)) -> c77(TOP(active(z0))) 50.15/13.80 K tuples:none 50.15/13.80 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 50.15/13.80 50.15/13.80 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 50.15/13.80 50.15/13.80 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 50.15/13.80 50.15/13.80 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 50.15/13.80 Removed 3 trailing nodes: 50.15/13.80 TOP(ok(z0)) -> c77(TOP(active(z0))) 50.15/13.80 PROPER(tt) -> c72 50.15/13.80 PROPER(0) -> c73 50.15/13.80 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (8) 50.15/13.80 Obligation: 50.15/13.80 Complexity Dependency Tuples Problem 50.15/13.80 50.15/13.80 Rules: 50.15/13.80 U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) 50.15/13.80 U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) 50.15/13.80 U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) 50.15/13.80 U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) 50.15/13.80 U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) 50.15/13.80 U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) 50.15/13.80 U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) 50.15/13.80 U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) 50.15/13.80 plus(mark(z0), z1) -> mark(plus(z0, z1)) 50.15/13.80 plus(z0, mark(z1)) -> mark(plus(z0, z1)) 50.15/13.80 plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) 50.15/13.80 x(mark(z0), z1) -> mark(x(z0, z1)) 50.15/13.80 x(z0, mark(z1)) -> mark(x(z0, z1)) 50.15/13.80 x(ok(z0), ok(z1)) -> ok(x(z0, z1)) 50.15/13.80 U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) 50.15/13.80 U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) 50.15/13.80 U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) 50.15/13.80 U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) 50.15/13.80 U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) 50.15/13.80 U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) 50.15/13.80 U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) 50.15/13.80 U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) 50.15/13.80 U15(mark(z0), z1) -> mark(U15(z0, z1)) 50.15/13.80 U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) 50.15/13.80 U16(mark(z0)) -> mark(U16(z0)) 50.15/13.80 U16(ok(z0)) -> ok(U16(z0)) 50.15/13.80 U21(mark(z0), z1) -> mark(U21(z0, z1)) 50.15/13.80 U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) 50.15/13.80 U22(mark(z0), z1) -> mark(U22(z0, z1)) 50.15/13.80 U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) 50.15/13.80 U23(mark(z0)) -> mark(U23(z0)) 50.15/13.80 U23(ok(z0)) -> ok(U23(z0)) 50.15/13.80 U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) 50.15/13.80 U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) 50.15/13.80 U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) 50.15/13.80 U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) 50.15/13.80 U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) 50.15/13.80 U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) 50.15/13.80 U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) 50.15/13.80 U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) 50.15/13.80 U35(mark(z0), z1) -> mark(U35(z0, z1)) 50.15/13.80 U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) 50.15/13.80 U36(mark(z0)) -> mark(U36(z0)) 50.15/13.80 U36(ok(z0)) -> ok(U36(z0)) 50.15/13.80 U41(mark(z0), z1) -> mark(U41(z0, z1)) 50.15/13.80 U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) 50.15/13.80 U42(mark(z0)) -> mark(U42(z0)) 50.15/13.80 U42(ok(z0)) -> ok(U42(z0)) 50.15/13.80 U51(mark(z0)) -> mark(U51(z0)) 50.15/13.80 U51(ok(z0)) -> ok(U51(z0)) 50.15/13.80 U61(mark(z0), z1) -> mark(U61(z0, z1)) 50.15/13.80 U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) 50.15/13.80 U62(mark(z0)) -> mark(U62(z0)) 50.15/13.80 U62(ok(z0)) -> ok(U62(z0)) 50.15/13.80 U71(mark(z0), z1) -> mark(U71(z0, z1)) 50.15/13.80 U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) 50.15/13.80 U72(mark(z0), z1) -> mark(U72(z0, z1)) 50.15/13.80 U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) 50.15/13.80 U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) 50.15/13.80 U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) 50.15/13.80 U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) 50.15/13.80 U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) 50.15/13.80 U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) 50.15/13.80 U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) 50.15/13.80 U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) 50.15/13.80 U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) 50.15/13.80 s(mark(z0)) -> mark(s(z0)) 50.15/13.80 s(ok(z0)) -> ok(s(z0)) 50.15/13.80 U91(mark(z0), z1) -> mark(U91(z0, z1)) 50.15/13.80 U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) 50.15/13.80 U92(mark(z0)) -> mark(U92(z0)) 50.15/13.80 U92(ok(z0)) -> ok(U92(z0)) 50.15/13.80 proper(tt) -> ok(tt) 50.15/13.80 proper(0) -> ok(0) 50.15/13.80 isNatKind(ok(z0)) -> ok(isNatKind(z0)) 50.15/13.80 isNat(ok(z0)) -> ok(isNat(z0)) 50.15/13.80 top(mark(z0)) -> top(proper(z0)) 50.15/13.80 top(ok(z0)) -> top(active(z0)) 50.15/13.80 Tuples: 50.15/13.80 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.80 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.80 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.80 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.80 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.80 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.80 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.80 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.80 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.80 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.80 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.80 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.80 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.80 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.80 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.80 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.80 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.80 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.80 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.80 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.80 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.80 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.80 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.80 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.80 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.80 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.80 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.80 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.80 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.80 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.80 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.80 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.80 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.80 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.80 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.80 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.80 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.80 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.80 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.80 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.80 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.80 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.80 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.80 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.80 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.80 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.80 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.80 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.80 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.80 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.80 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.80 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.80 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.80 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.80 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.80 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.80 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.80 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.80 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.80 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.80 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.80 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.80 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.80 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.80 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.80 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.80 S(mark(z0)) -> c66(S(z0)) 50.15/13.80 S(ok(z0)) -> c67(S(z0)) 50.15/13.80 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.80 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.80 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.80 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.80 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.80 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.80 TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) 50.15/13.80 S tuples: 50.15/13.80 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.80 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.80 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.80 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.80 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.80 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.80 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.80 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.80 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.80 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.80 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.80 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.80 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.80 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.80 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.80 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.80 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.80 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.80 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.80 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.80 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.80 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.80 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.80 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.80 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.80 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.80 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.80 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.80 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.80 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.80 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.80 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.80 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.80 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.80 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.80 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.80 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.80 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.80 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.80 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.80 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.80 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.80 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.80 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.80 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.80 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.80 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.80 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.80 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.80 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.80 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.80 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.80 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.80 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.80 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.80 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.80 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.80 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.80 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.80 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.80 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.80 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.80 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.80 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.80 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.80 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.80 S(mark(z0)) -> c66(S(z0)) 50.15/13.80 S(ok(z0)) -> c67(S(z0)) 50.15/13.80 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.80 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.80 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.80 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.80 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.80 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.80 TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) 50.15/13.80 K tuples:none 50.15/13.80 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 50.15/13.80 50.15/13.80 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 50.15/13.80 50.15/13.80 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 50.15/13.80 50.15/13.80 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 50.15/13.80 Removed 1 trailing tuple parts 50.15/13.80 ---------------------------------------- 50.15/13.80 50.15/13.80 (10) 50.15/13.80 Obligation: 50.15/13.80 Complexity Dependency Tuples Problem 50.15/13.80 50.15/13.80 Rules: 50.15/13.80 U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) 50.15/13.80 U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) 50.15/13.80 U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) 50.15/13.80 U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) 50.15/13.80 U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) 50.15/13.80 U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) 50.15/13.80 U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) 50.15/13.80 U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) 50.15/13.80 plus(mark(z0), z1) -> mark(plus(z0, z1)) 50.15/13.80 plus(z0, mark(z1)) -> mark(plus(z0, z1)) 50.15/13.80 plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) 50.15/13.80 x(mark(z0), z1) -> mark(x(z0, z1)) 50.15/13.80 x(z0, mark(z1)) -> mark(x(z0, z1)) 50.15/13.80 x(ok(z0), ok(z1)) -> ok(x(z0, z1)) 50.15/13.80 U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) 50.15/13.80 U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) 50.15/13.80 U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) 50.15/13.80 U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) 50.15/13.80 U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) 50.15/13.81 U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) 50.15/13.81 U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) 50.15/13.81 U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) 50.15/13.81 U15(mark(z0), z1) -> mark(U15(z0, z1)) 50.15/13.81 U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) 50.15/13.81 U16(mark(z0)) -> mark(U16(z0)) 50.15/13.81 U16(ok(z0)) -> ok(U16(z0)) 50.15/13.81 U21(mark(z0), z1) -> mark(U21(z0, z1)) 50.15/13.81 U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) 50.15/13.81 U22(mark(z0), z1) -> mark(U22(z0, z1)) 50.15/13.81 U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) 50.15/13.81 U23(mark(z0)) -> mark(U23(z0)) 50.15/13.81 U23(ok(z0)) -> ok(U23(z0)) 50.15/13.81 U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) 50.15/13.81 U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) 50.15/13.81 U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) 50.15/13.81 U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) 50.15/13.81 U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) 50.15/13.81 U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) 50.15/13.81 U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) 50.15/13.81 U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) 50.15/13.81 U35(mark(z0), z1) -> mark(U35(z0, z1)) 50.15/13.81 U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) 50.15/13.81 U36(mark(z0)) -> mark(U36(z0)) 50.15/13.81 U36(ok(z0)) -> ok(U36(z0)) 50.15/13.81 U41(mark(z0), z1) -> mark(U41(z0, z1)) 50.15/13.81 U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) 50.15/13.81 U42(mark(z0)) -> mark(U42(z0)) 50.15/13.81 U42(ok(z0)) -> ok(U42(z0)) 50.15/13.81 U51(mark(z0)) -> mark(U51(z0)) 50.15/13.81 U51(ok(z0)) -> ok(U51(z0)) 50.15/13.81 U61(mark(z0), z1) -> mark(U61(z0, z1)) 50.15/13.81 U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) 50.15/13.81 U62(mark(z0)) -> mark(U62(z0)) 50.15/13.81 U62(ok(z0)) -> ok(U62(z0)) 50.15/13.81 U71(mark(z0), z1) -> mark(U71(z0, z1)) 50.15/13.81 U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) 50.15/13.81 U72(mark(z0), z1) -> mark(U72(z0, z1)) 50.15/13.81 U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) 50.15/13.81 U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) 50.15/13.81 U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) 50.15/13.81 U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) 50.15/13.81 U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) 50.15/13.81 U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) 50.15/13.81 U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) 50.15/13.81 U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) 50.15/13.81 U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) 50.15/13.81 s(mark(z0)) -> mark(s(z0)) 50.15/13.81 s(ok(z0)) -> ok(s(z0)) 50.15/13.81 U91(mark(z0), z1) -> mark(U91(z0, z1)) 50.15/13.81 U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) 50.15/13.81 U92(mark(z0)) -> mark(U92(z0)) 50.15/13.81 U92(ok(z0)) -> ok(U92(z0)) 50.15/13.81 proper(tt) -> ok(tt) 50.15/13.81 proper(0) -> ok(0) 50.15/13.81 isNatKind(ok(z0)) -> ok(isNatKind(z0)) 50.15/13.81 isNat(ok(z0)) -> ok(isNat(z0)) 50.15/13.81 top(mark(z0)) -> top(proper(z0)) 50.15/13.81 top(ok(z0)) -> top(active(z0)) 50.15/13.81 Tuples: 50.15/13.81 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.81 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.81 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.81 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.81 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.81 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.81 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.81 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.81 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.81 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.81 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.81 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.81 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.81 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.81 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.81 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.81 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.81 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.81 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.81 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.81 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.81 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.81 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.81 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.81 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.81 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.81 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.81 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.81 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.81 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.81 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.81 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.81 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.81 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.81 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.81 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.81 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.81 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.81 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.81 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.81 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.81 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.81 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.81 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.81 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.81 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.81 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.81 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.81 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.81 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.81 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.81 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.81 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.81 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.81 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.81 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.81 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.81 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.81 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.81 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.81 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.81 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.81 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.81 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.81 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.81 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.81 S(mark(z0)) -> c66(S(z0)) 50.15/13.81 S(ok(z0)) -> c67(S(z0)) 50.15/13.81 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.81 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.81 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.81 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.81 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.81 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.81 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.15/13.81 S tuples: 50.15/13.81 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.81 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.81 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.81 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.81 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.81 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.81 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.81 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.81 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.81 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.81 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.81 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.81 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.81 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.81 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.81 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.81 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.81 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.81 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.81 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.81 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.81 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.81 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.81 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.81 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.81 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.81 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.81 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.81 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.81 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.81 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.81 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.81 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.81 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.81 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.81 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.81 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.81 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.81 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.81 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.81 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.81 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.81 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.81 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.81 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.81 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.81 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.81 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.81 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.81 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.82 S(mark(z0)) -> c66(S(z0)) 50.15/13.82 S(ok(z0)) -> c67(S(z0)) 50.15/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.15/13.82 K tuples:none 50.15/13.82 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 50.15/13.82 50.15/13.82 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 50.15/13.82 50.15/13.82 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 50.15/13.82 50.15/13.82 50.15/13.82 ---------------------------------------- 50.15/13.82 50.15/13.82 (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) 50.15/13.82 The following rules are not usable and were removed: 50.15/13.82 U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) 50.15/13.82 U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) 50.15/13.82 U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) 50.15/13.82 U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) 50.15/13.82 U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) 50.15/13.82 U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) 50.15/13.82 U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) 50.15/13.82 U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) 50.15/13.82 plus(mark(z0), z1) -> mark(plus(z0, z1)) 50.15/13.82 plus(z0, mark(z1)) -> mark(plus(z0, z1)) 50.15/13.82 plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) 50.15/13.82 x(mark(z0), z1) -> mark(x(z0, z1)) 50.15/13.82 x(z0, mark(z1)) -> mark(x(z0, z1)) 50.15/13.82 x(ok(z0), ok(z1)) -> ok(x(z0, z1)) 50.15/13.82 U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) 50.15/13.82 U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) 50.15/13.82 U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) 50.15/13.82 U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) 50.15/13.82 U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) 50.15/13.82 U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) 50.15/13.82 U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) 50.15/13.82 U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) 50.15/13.82 U15(mark(z0), z1) -> mark(U15(z0, z1)) 50.15/13.82 U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) 50.15/13.82 U16(mark(z0)) -> mark(U16(z0)) 50.15/13.82 U16(ok(z0)) -> ok(U16(z0)) 50.15/13.82 U21(mark(z0), z1) -> mark(U21(z0, z1)) 50.15/13.82 U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) 50.15/13.82 U22(mark(z0), z1) -> mark(U22(z0, z1)) 50.15/13.82 U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) 50.15/13.82 U23(mark(z0)) -> mark(U23(z0)) 50.15/13.82 U23(ok(z0)) -> ok(U23(z0)) 50.15/13.82 U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) 50.15/13.82 U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) 50.15/13.82 U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) 50.15/13.82 U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) 50.15/13.82 U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) 50.15/13.82 U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) 50.15/13.82 U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) 50.15/13.82 U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) 50.15/13.82 U35(mark(z0), z1) -> mark(U35(z0, z1)) 50.15/13.82 U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) 50.15/13.82 U36(mark(z0)) -> mark(U36(z0)) 50.15/13.82 U36(ok(z0)) -> ok(U36(z0)) 50.15/13.82 U41(mark(z0), z1) -> mark(U41(z0, z1)) 50.15/13.82 U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) 50.15/13.82 U42(mark(z0)) -> mark(U42(z0)) 50.15/13.82 U42(ok(z0)) -> ok(U42(z0)) 50.15/13.82 U51(mark(z0)) -> mark(U51(z0)) 50.15/13.82 U51(ok(z0)) -> ok(U51(z0)) 50.15/13.82 U61(mark(z0), z1) -> mark(U61(z0, z1)) 50.15/13.82 U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) 50.15/13.82 U62(mark(z0)) -> mark(U62(z0)) 50.15/13.82 U62(ok(z0)) -> ok(U62(z0)) 50.15/13.82 U71(mark(z0), z1) -> mark(U71(z0, z1)) 50.15/13.82 U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) 50.15/13.82 U72(mark(z0), z1) -> mark(U72(z0, z1)) 50.15/13.82 U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) 50.15/13.82 U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) 50.15/13.82 U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) 50.15/13.82 U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) 50.15/13.82 U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) 50.15/13.82 U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) 50.15/13.82 U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) 50.15/13.82 U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) 50.15/13.82 U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) 50.15/13.82 s(mark(z0)) -> mark(s(z0)) 50.15/13.82 s(ok(z0)) -> ok(s(z0)) 50.15/13.82 U91(mark(z0), z1) -> mark(U91(z0, z1)) 50.15/13.82 U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) 50.15/13.82 U92(mark(z0)) -> mark(U92(z0)) 50.15/13.82 U92(ok(z0)) -> ok(U92(z0)) 50.15/13.82 isNatKind(ok(z0)) -> ok(isNatKind(z0)) 50.15/13.82 isNat(ok(z0)) -> ok(isNat(z0)) 50.15/13.82 top(mark(z0)) -> top(proper(z0)) 50.15/13.82 top(ok(z0)) -> top(active(z0)) 50.15/13.82 50.15/13.82 ---------------------------------------- 50.15/13.82 50.15/13.82 (12) 50.15/13.82 Obligation: 50.15/13.82 Complexity Dependency Tuples Problem 50.15/13.82 50.15/13.82 Rules: 50.15/13.82 proper(tt) -> ok(tt) 50.15/13.82 proper(0) -> ok(0) 50.15/13.82 Tuples: 50.15/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.82 S(mark(z0)) -> c66(S(z0)) 50.15/13.82 S(ok(z0)) -> c67(S(z0)) 50.15/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.15/13.82 S tuples: 50.15/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.82 S(mark(z0)) -> c66(S(z0)) 50.15/13.82 S(ok(z0)) -> c67(S(z0)) 50.15/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.15/13.82 K tuples:none 50.15/13.82 Defined Rule Symbols: proper_1 50.15/13.82 50.15/13.82 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 50.15/13.82 50.15/13.82 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 50.15/13.82 50.15/13.82 50.15/13.82 ---------------------------------------- 50.15/13.82 50.15/13.82 (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) 50.15/13.82 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 50.15/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.82 S(mark(z0)) -> c66(S(z0)) 50.15/13.82 S(ok(z0)) -> c67(S(z0)) 50.15/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.82 We considered the (Usable) Rules: 50.15/13.82 proper(tt) -> ok(tt) 50.15/13.82 proper(0) -> ok(0) 50.15/13.82 And the Tuples: 50.15/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.82 S(mark(z0)) -> c66(S(z0)) 50.15/13.82 S(ok(z0)) -> c67(S(z0)) 50.15/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.15/13.82 The order we found is given by the following interpretation: 50.15/13.82 50.15/13.82 Polynomial interpretation : 50.15/13.82 50.15/13.82 POL(0) = [1] 50.15/13.82 POL(ISNAT(x_1)) = x_1 50.15/13.82 POL(ISNATKIND(x_1)) = x_1 50.15/13.82 POL(PLUS(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(S(x_1)) = x_1 50.15/13.82 POL(TOP(x_1)) = x_1 50.15/13.82 POL(U101'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U102'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U103'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U104'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U11'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U12'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U13'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U14'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U15'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U16'(x_1)) = x_1 50.15/13.82 POL(U21'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U22'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U23'(x_1)) = x_1 50.15/13.82 POL(U31'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U32'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U33'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U34'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U35'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U36'(x_1)) = x_1 50.15/13.82 POL(U41'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U42'(x_1)) = x_1 50.15/13.82 POL(U51'(x_1)) = x_1 50.15/13.82 POL(U61'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U62'(x_1)) = x_1 50.15/13.82 POL(U71'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U72'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U81'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U82'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U83'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U84'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 50.15/13.82 POL(U91'(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(U92'(x_1)) = x_1 50.15/13.82 POL(X(x_1, x_2)) = x_1 + x_2 50.15/13.82 POL(c(x_1)) = x_1 50.15/13.82 POL(c1(x_1)) = x_1 50.15/13.82 POL(c10(x_1)) = x_1 50.15/13.82 POL(c11(x_1)) = x_1 50.15/13.82 POL(c12(x_1)) = x_1 50.15/13.82 POL(c13(x_1)) = x_1 50.15/13.82 POL(c14(x_1)) = x_1 50.15/13.82 POL(c15(x_1)) = x_1 50.15/13.82 POL(c16(x_1)) = x_1 50.15/13.82 POL(c17(x_1)) = x_1 50.15/13.82 POL(c18(x_1)) = x_1 50.15/13.82 POL(c19(x_1)) = x_1 50.15/13.82 POL(c2(x_1)) = x_1 50.15/13.82 POL(c20(x_1)) = x_1 50.15/13.82 POL(c21(x_1)) = x_1 50.15/13.82 POL(c22(x_1)) = x_1 50.15/13.82 POL(c23(x_1)) = x_1 50.15/13.82 POL(c24(x_1)) = x_1 50.15/13.82 POL(c25(x_1)) = x_1 50.15/13.82 POL(c26(x_1)) = x_1 50.15/13.82 POL(c27(x_1)) = x_1 50.15/13.82 POL(c28(x_1)) = x_1 50.15/13.82 POL(c29(x_1)) = x_1 50.15/13.82 POL(c3(x_1)) = x_1 50.15/13.82 POL(c30(x_1)) = x_1 50.15/13.82 POL(c31(x_1)) = x_1 50.15/13.82 POL(c32(x_1)) = x_1 50.15/13.82 POL(c33(x_1)) = x_1 50.15/13.82 POL(c34(x_1)) = x_1 50.15/13.82 POL(c35(x_1)) = x_1 50.15/13.82 POL(c36(x_1)) = x_1 50.15/13.82 POL(c37(x_1)) = x_1 50.15/13.82 POL(c38(x_1)) = x_1 50.15/13.82 POL(c39(x_1)) = x_1 50.15/13.82 POL(c4(x_1)) = x_1 50.15/13.82 POL(c40(x_1)) = x_1 50.15/13.82 POL(c41(x_1)) = x_1 50.15/13.82 POL(c42(x_1)) = x_1 50.15/13.82 POL(c43(x_1)) = x_1 50.15/13.82 POL(c44(x_1)) = x_1 50.15/13.82 POL(c45(x_1)) = x_1 50.15/13.82 POL(c46(x_1)) = x_1 50.15/13.82 POL(c47(x_1)) = x_1 50.15/13.82 POL(c48(x_1)) = x_1 50.15/13.82 POL(c49(x_1)) = x_1 50.15/13.82 POL(c5(x_1)) = x_1 50.15/13.82 POL(c50(x_1)) = x_1 50.15/13.82 POL(c51(x_1)) = x_1 50.15/13.82 POL(c52(x_1)) = x_1 50.15/13.82 POL(c53(x_1)) = x_1 50.15/13.82 POL(c54(x_1)) = x_1 50.15/13.82 POL(c55(x_1)) = x_1 50.15/13.82 POL(c56(x_1)) = x_1 50.15/13.82 POL(c57(x_1)) = x_1 50.15/13.82 POL(c58(x_1)) = x_1 50.15/13.82 POL(c59(x_1)) = x_1 50.15/13.82 POL(c6(x_1)) = x_1 50.15/13.82 POL(c60(x_1)) = x_1 50.15/13.82 POL(c61(x_1)) = x_1 50.15/13.82 POL(c62(x_1)) = x_1 50.15/13.82 POL(c63(x_1)) = x_1 50.15/13.82 POL(c64(x_1)) = x_1 50.15/13.82 POL(c65(x_1)) = x_1 50.15/13.82 POL(c66(x_1)) = x_1 50.15/13.82 POL(c67(x_1)) = x_1 50.15/13.82 POL(c68(x_1)) = x_1 50.15/13.82 POL(c69(x_1)) = x_1 50.15/13.82 POL(c7(x_1)) = x_1 50.15/13.82 POL(c70(x_1)) = x_1 50.15/13.82 POL(c71(x_1)) = x_1 50.15/13.82 POL(c74(x_1)) = x_1 50.15/13.82 POL(c75(x_1)) = x_1 50.15/13.82 POL(c76(x_1)) = x_1 50.15/13.82 POL(c8(x_1)) = x_1 50.15/13.82 POL(c9(x_1)) = x_1 50.15/13.82 POL(mark(x_1)) = [1] + x_1 50.15/13.82 POL(ok(x_1)) = [1] + x_1 50.15/13.82 POL(proper(x_1)) = [1] + x_1 50.15/13.82 POL(tt) = [1] 50.15/13.82 50.15/13.82 ---------------------------------------- 50.15/13.82 50.15/13.82 (14) 50.15/13.82 Obligation: 50.15/13.82 Complexity Dependency Tuples Problem 50.15/13.82 50.15/13.82 Rules: 50.15/13.82 proper(tt) -> ok(tt) 50.15/13.82 proper(0) -> ok(0) 50.15/13.82 Tuples: 50.15/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.15/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.15/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.15/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.15/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.15/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.15/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.15/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.15/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.15/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.15/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.15/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.15/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.15/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.15/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.15/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.15/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.15/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.15/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.15/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.15/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.15/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.15/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.15/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.15/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.15/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.15/13.82 S(mark(z0)) -> c66(S(z0)) 50.15/13.82 S(ok(z0)) -> c67(S(z0)) 50.15/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.15/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.15/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.15/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.15/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.15/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.15/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.15/13.82 S tuples: 50.15/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.15/13.82 K tuples: 50.15/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.15/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.15/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.15/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.15/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.15/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.15/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.15/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.15/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.15/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.15/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.15/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.15/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.15/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.15/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.15/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.15/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.15/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.15/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.15/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.15/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.15/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.15/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.15/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.15/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.15/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.15/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.15/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.15/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.15/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.15/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.15/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.15/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.15/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.15/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.15/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.15/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.15/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.15/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.15/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.15/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.42/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.42/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.42/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.42/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.42/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.42/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.42/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.42/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.42/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.42/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.42/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.42/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.42/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.42/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.42/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.42/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.42/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.42/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.42/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.42/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.42/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.42/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.42/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.42/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.42/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.42/13.82 S(mark(z0)) -> c66(S(z0)) 50.42/13.82 S(ok(z0)) -> c67(S(z0)) 50.42/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.42/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.42/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.42/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.42/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.42/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.42/13.82 Defined Rule Symbols: proper_1 50.42/13.82 50.42/13.82 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 50.42/13.82 50.42/13.82 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 50.42/13.82 50.42/13.82 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) 50.42/13.82 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 50.42/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.42/13.82 We considered the (Usable) Rules: 50.42/13.82 proper(tt) -> ok(tt) 50.42/13.82 proper(0) -> ok(0) 50.42/13.82 And the Tuples: 50.42/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.42/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.42/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.42/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.42/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.42/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.42/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.42/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.42/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.42/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.42/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.42/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.42/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.42/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.42/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.42/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.42/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.42/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.42/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.42/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.42/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.42/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.42/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.42/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.42/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.42/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.42/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.42/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.42/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.42/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.42/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.42/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.42/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.42/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.42/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.42/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.42/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.42/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.42/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.42/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.42/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.42/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.42/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.42/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.42/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.42/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.42/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.42/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.42/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.42/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.42/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.42/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.42/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.42/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.42/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.42/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.42/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.42/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.42/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.42/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.42/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.42/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.42/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.42/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.42/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.42/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.42/13.82 S(mark(z0)) -> c66(S(z0)) 50.42/13.82 S(ok(z0)) -> c67(S(z0)) 50.42/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.42/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.42/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.42/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.42/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.42/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.42/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.42/13.82 The order we found is given by the following interpretation: 50.42/13.82 50.42/13.82 Polynomial interpretation : 50.42/13.82 50.42/13.82 POL(0) = [1] 50.42/13.82 POL(ISNAT(x_1)) = 0 50.42/13.82 POL(ISNATKIND(x_1)) = 0 50.42/13.82 POL(PLUS(x_1, x_2)) = 0 50.42/13.82 POL(S(x_1)) = 0 50.42/13.82 POL(TOP(x_1)) = x_1 50.42/13.82 POL(U101'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U102'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U103'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U104'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U11'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U12'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U13'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U14'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U15'(x_1, x_2)) = 0 50.42/13.82 POL(U16'(x_1)) = 0 50.42/13.82 POL(U21'(x_1, x_2)) = 0 50.42/13.82 POL(U22'(x_1, x_2)) = 0 50.42/13.82 POL(U23'(x_1)) = 0 50.42/13.82 POL(U31'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U32'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U33'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U34'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U35'(x_1, x_2)) = 0 50.42/13.82 POL(U36'(x_1)) = 0 50.42/13.82 POL(U41'(x_1, x_2)) = 0 50.42/13.82 POL(U42'(x_1)) = 0 50.42/13.82 POL(U51'(x_1)) = 0 50.42/13.82 POL(U61'(x_1, x_2)) = 0 50.42/13.82 POL(U62'(x_1)) = 0 50.42/13.82 POL(U71'(x_1, x_2)) = 0 50.42/13.82 POL(U72'(x_1, x_2)) = 0 50.42/13.82 POL(U81'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U82'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U83'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U84'(x_1, x_2, x_3)) = 0 50.42/13.82 POL(U91'(x_1, x_2)) = 0 50.42/13.82 POL(U92'(x_1)) = 0 50.42/13.82 POL(X(x_1, x_2)) = 0 50.42/13.82 POL(c(x_1)) = x_1 50.42/13.82 POL(c1(x_1)) = x_1 50.42/13.82 POL(c10(x_1)) = x_1 50.42/13.82 POL(c11(x_1)) = x_1 50.42/13.82 POL(c12(x_1)) = x_1 50.42/13.82 POL(c13(x_1)) = x_1 50.42/13.82 POL(c14(x_1)) = x_1 50.42/13.82 POL(c15(x_1)) = x_1 50.42/13.82 POL(c16(x_1)) = x_1 50.42/13.82 POL(c17(x_1)) = x_1 50.42/13.82 POL(c18(x_1)) = x_1 50.42/13.82 POL(c19(x_1)) = x_1 50.42/13.82 POL(c2(x_1)) = x_1 50.42/13.82 POL(c20(x_1)) = x_1 50.42/13.82 POL(c21(x_1)) = x_1 50.42/13.82 POL(c22(x_1)) = x_1 50.42/13.82 POL(c23(x_1)) = x_1 50.42/13.82 POL(c24(x_1)) = x_1 50.42/13.82 POL(c25(x_1)) = x_1 50.42/13.82 POL(c26(x_1)) = x_1 50.42/13.82 POL(c27(x_1)) = x_1 50.42/13.82 POL(c28(x_1)) = x_1 50.42/13.82 POL(c29(x_1)) = x_1 50.42/13.82 POL(c3(x_1)) = x_1 50.42/13.82 POL(c30(x_1)) = x_1 50.42/13.82 POL(c31(x_1)) = x_1 50.42/13.82 POL(c32(x_1)) = x_1 50.42/13.82 POL(c33(x_1)) = x_1 50.42/13.82 POL(c34(x_1)) = x_1 50.42/13.82 POL(c35(x_1)) = x_1 50.42/13.82 POL(c36(x_1)) = x_1 50.42/13.82 POL(c37(x_1)) = x_1 50.42/13.82 POL(c38(x_1)) = x_1 50.42/13.82 POL(c39(x_1)) = x_1 50.42/13.82 POL(c4(x_1)) = x_1 50.42/13.82 POL(c40(x_1)) = x_1 50.42/13.82 POL(c41(x_1)) = x_1 50.42/13.82 POL(c42(x_1)) = x_1 50.42/13.82 POL(c43(x_1)) = x_1 50.42/13.82 POL(c44(x_1)) = x_1 50.42/13.82 POL(c45(x_1)) = x_1 50.42/13.82 POL(c46(x_1)) = x_1 50.42/13.82 POL(c47(x_1)) = x_1 50.42/13.82 POL(c48(x_1)) = x_1 50.42/13.82 POL(c49(x_1)) = x_1 50.42/13.82 POL(c5(x_1)) = x_1 50.42/13.82 POL(c50(x_1)) = x_1 50.42/13.82 POL(c51(x_1)) = x_1 50.42/13.82 POL(c52(x_1)) = x_1 50.42/13.82 POL(c53(x_1)) = x_1 50.42/13.82 POL(c54(x_1)) = x_1 50.42/13.82 POL(c55(x_1)) = x_1 50.42/13.82 POL(c56(x_1)) = x_1 50.42/13.82 POL(c57(x_1)) = x_1 50.42/13.82 POL(c58(x_1)) = x_1 50.42/13.82 POL(c59(x_1)) = x_1 50.42/13.82 POL(c6(x_1)) = x_1 50.42/13.82 POL(c60(x_1)) = x_1 50.42/13.82 POL(c61(x_1)) = x_1 50.42/13.82 POL(c62(x_1)) = x_1 50.42/13.82 POL(c63(x_1)) = x_1 50.42/13.82 POL(c64(x_1)) = x_1 50.42/13.82 POL(c65(x_1)) = x_1 50.42/13.82 POL(c66(x_1)) = x_1 50.42/13.82 POL(c67(x_1)) = x_1 50.42/13.82 POL(c68(x_1)) = x_1 50.42/13.82 POL(c69(x_1)) = x_1 50.42/13.82 POL(c7(x_1)) = x_1 50.42/13.82 POL(c70(x_1)) = x_1 50.42/13.82 POL(c71(x_1)) = x_1 50.42/13.82 POL(c74(x_1)) = x_1 50.42/13.82 POL(c75(x_1)) = x_1 50.42/13.82 POL(c76(x_1)) = x_1 50.42/13.82 POL(c8(x_1)) = x_1 50.42/13.82 POL(c9(x_1)) = x_1 50.42/13.82 POL(mark(x_1)) = [1] + x_1 50.42/13.82 POL(ok(x_1)) = [1] 50.42/13.82 POL(proper(x_1)) = x_1 50.42/13.82 POL(tt) = [1] 50.42/13.82 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (16) 50.42/13.82 Obligation: 50.42/13.82 Complexity Dependency Tuples Problem 50.42/13.82 50.42/13.82 Rules: 50.42/13.82 proper(tt) -> ok(tt) 50.42/13.82 proper(0) -> ok(0) 50.42/13.82 Tuples: 50.42/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.42/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.42/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.42/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.42/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.42/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.42/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.42/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.42/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.42/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.42/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.42/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.42/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.42/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.42/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.42/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.42/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.42/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.42/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.42/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.42/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.42/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.42/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.42/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.42/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.42/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.42/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.42/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.42/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.42/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.42/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.42/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.42/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.42/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.42/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.42/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.42/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.42/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.42/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.42/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.42/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.42/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.42/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.42/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.42/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.42/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.42/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.42/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.42/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.42/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.42/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.42/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.42/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.42/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.42/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.42/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.42/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.42/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.42/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.42/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.42/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.42/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.42/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.42/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.42/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.42/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.42/13.82 S(mark(z0)) -> c66(S(z0)) 50.42/13.82 S(ok(z0)) -> c67(S(z0)) 50.42/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.42/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.42/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.42/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.42/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.42/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.42/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.42/13.82 S tuples:none 50.42/13.82 K tuples: 50.42/13.82 U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) 50.42/13.82 U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) 50.42/13.82 U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) 50.42/13.82 U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) 50.42/13.82 U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) 50.42/13.82 U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) 50.42/13.82 U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) 50.42/13.82 U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) 50.42/13.82 PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) 50.42/13.82 PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) 50.42/13.82 PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) 50.42/13.82 X(mark(z0), z1) -> c11(X(z0, z1)) 50.42/13.82 X(z0, mark(z1)) -> c12(X(z0, z1)) 50.42/13.82 X(ok(z0), ok(z1)) -> c13(X(z0, z1)) 50.42/13.82 U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) 50.42/13.82 U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) 50.42/13.82 U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) 50.42/13.82 U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) 50.42/13.82 U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) 50.42/13.82 U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) 50.42/13.82 U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) 50.42/13.82 U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) 50.42/13.82 U15'(mark(z0), z1) -> c22(U15'(z0, z1)) 50.42/13.82 U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) 50.42/13.82 U16'(mark(z0)) -> c24(U16'(z0)) 50.42/13.82 U16'(ok(z0)) -> c25(U16'(z0)) 50.42/13.82 U21'(mark(z0), z1) -> c26(U21'(z0, z1)) 50.42/13.82 U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) 50.42/13.82 U22'(mark(z0), z1) -> c28(U22'(z0, z1)) 50.42/13.82 U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) 50.42/13.82 U23'(mark(z0)) -> c30(U23'(z0)) 50.42/13.82 U23'(ok(z0)) -> c31(U23'(z0)) 50.42/13.82 U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) 50.42/13.82 U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) 50.42/13.82 U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) 50.42/13.82 U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) 50.42/13.82 U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) 50.42/13.82 U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) 50.42/13.82 U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) 50.42/13.82 U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) 50.42/13.82 U35'(mark(z0), z1) -> c40(U35'(z0, z1)) 50.42/13.82 U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) 50.42/13.82 U36'(mark(z0)) -> c42(U36'(z0)) 50.42/13.82 U36'(ok(z0)) -> c43(U36'(z0)) 50.42/13.82 U41'(mark(z0), z1) -> c44(U41'(z0, z1)) 50.42/13.82 U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) 50.42/13.82 U42'(mark(z0)) -> c46(U42'(z0)) 50.42/13.82 U42'(ok(z0)) -> c47(U42'(z0)) 50.42/13.82 U51'(mark(z0)) -> c48(U51'(z0)) 50.42/13.82 U51'(ok(z0)) -> c49(U51'(z0)) 50.42/13.82 U61'(mark(z0), z1) -> c50(U61'(z0, z1)) 50.42/13.82 U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) 50.42/13.82 U62'(mark(z0)) -> c52(U62'(z0)) 50.42/13.82 U62'(ok(z0)) -> c53(U62'(z0)) 50.42/13.82 U71'(mark(z0), z1) -> c54(U71'(z0, z1)) 50.42/13.82 U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) 50.42/13.82 U72'(mark(z0), z1) -> c56(U72'(z0, z1)) 50.42/13.82 U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) 50.42/13.82 U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) 50.42/13.82 U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) 50.42/13.82 U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) 50.42/13.82 U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) 50.42/13.82 U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) 50.42/13.82 U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) 50.42/13.82 U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) 50.42/13.82 U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) 50.42/13.82 S(mark(z0)) -> c66(S(z0)) 50.42/13.82 S(ok(z0)) -> c67(S(z0)) 50.42/13.82 U91'(mark(z0), z1) -> c68(U91'(z0, z1)) 50.42/13.82 U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) 50.42/13.82 U92'(mark(z0)) -> c70(U92'(z0)) 50.42/13.82 U92'(ok(z0)) -> c71(U92'(z0)) 50.42/13.82 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) 50.42/13.82 ISNAT(ok(z0)) -> c75(ISNAT(z0)) 50.42/13.82 TOP(mark(z0)) -> c76(TOP(proper(z0))) 50.42/13.82 Defined Rule Symbols: proper_1 50.42/13.82 50.42/13.82 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 50.42/13.82 50.42/13.82 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 50.42/13.82 50.42/13.82 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (17) SIsEmptyProof (BOTH BOUNDS(ID, ID)) 50.42/13.82 The set S is empty 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (18) 50.42/13.82 BOUNDS(1, 1) 50.42/13.82 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (19) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 50.42/13.82 Transformed a relative TRS into a decreasing-loop problem. 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (20) 50.42/13.82 Obligation: 50.42/13.82 Analyzing the following TRS for decreasing loops: 50.42/13.82 50.42/13.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 50.42/13.82 50.42/13.82 50.42/13.82 The TRS R consists of the following rules: 50.42/13.82 50.42/13.82 active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) 50.42/13.82 active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) 50.42/13.82 active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) 50.42/13.82 active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) 50.42/13.82 active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) 50.42/13.82 active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) 50.42/13.82 active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) 50.42/13.82 active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) 50.42/13.82 active(U15(tt, V2)) -> mark(U16(isNat(V2))) 50.42/13.82 active(U16(tt)) -> mark(tt) 50.42/13.82 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 50.42/13.82 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 50.42/13.82 active(U23(tt)) -> mark(tt) 50.42/13.82 active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) 50.42/13.82 active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) 50.42/13.82 active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) 50.42/13.82 active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) 50.42/13.82 active(U35(tt, V2)) -> mark(U36(isNat(V2))) 50.42/13.82 active(U36(tt)) -> mark(tt) 50.42/13.82 active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) 50.42/13.82 active(U42(tt)) -> mark(tt) 50.42/13.82 active(U51(tt)) -> mark(tt) 50.42/13.82 active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) 50.42/13.82 active(U62(tt)) -> mark(tt) 50.42/13.82 active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) 50.42/13.82 active(U72(tt, N)) -> mark(N) 50.42/13.82 active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) 50.42/13.82 active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) 50.42/13.82 active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) 50.42/13.82 active(U84(tt, M, N)) -> mark(s(plus(N, M))) 50.42/13.82 active(U91(tt, N)) -> mark(U92(isNatKind(N))) 50.42/13.82 active(U92(tt)) -> mark(0) 50.42/13.82 active(isNat(0)) -> mark(tt) 50.42/13.82 active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) 50.42/13.82 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 50.42/13.82 active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) 50.42/13.82 active(isNatKind(0)) -> mark(tt) 50.42/13.82 active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) 50.42/13.82 active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) 50.42/13.82 active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 50.42/13.82 active(plus(N, 0)) -> mark(U71(isNat(N), N)) 50.42/13.82 active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) 50.42/13.82 active(x(N, 0)) -> mark(U91(isNat(N), N)) 50.42/13.82 active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 50.42/13.82 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 50.42/13.82 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 50.42/13.82 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 50.42/13.82 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 50.42/13.82 active(plus(X1, X2)) -> plus(active(X1), X2) 50.42/13.82 active(plus(X1, X2)) -> plus(X1, active(X2)) 50.42/13.82 active(x(X1, X2)) -> x(active(X1), X2) 50.42/13.82 active(x(X1, X2)) -> x(X1, active(X2)) 50.42/13.82 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 50.42/13.82 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 50.42/13.82 active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) 50.42/13.82 active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) 50.42/13.82 active(U15(X1, X2)) -> U15(active(X1), X2) 50.42/13.82 active(U16(X)) -> U16(active(X)) 50.42/13.82 active(U21(X1, X2)) -> U21(active(X1), X2) 50.42/13.82 active(U22(X1, X2)) -> U22(active(X1), X2) 50.42/13.82 active(U23(X)) -> U23(active(X)) 50.42/13.82 active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) 50.42/13.82 active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) 50.42/13.82 active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) 50.42/13.82 active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) 50.42/13.82 active(U35(X1, X2)) -> U35(active(X1), X2) 50.42/13.82 active(U36(X)) -> U36(active(X)) 50.42/13.82 active(U41(X1, X2)) -> U41(active(X1), X2) 50.42/13.82 active(U42(X)) -> U42(active(X)) 50.42/13.82 active(U51(X)) -> U51(active(X)) 50.42/13.82 active(U61(X1, X2)) -> U61(active(X1), X2) 50.42/13.82 active(U62(X)) -> U62(active(X)) 50.42/13.82 active(U71(X1, X2)) -> U71(active(X1), X2) 50.42/13.82 active(U72(X1, X2)) -> U72(active(X1), X2) 50.42/13.82 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 50.42/13.82 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 50.42/13.82 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 50.42/13.82 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 50.42/13.82 active(s(X)) -> s(active(X)) 50.42/13.82 active(U91(X1, X2)) -> U91(active(X1), X2) 50.42/13.82 active(U92(X)) -> U92(active(X)) 50.42/13.82 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 50.42/13.82 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 50.42/13.82 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 50.42/13.82 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 50.42/13.82 plus(mark(X1), X2) -> mark(plus(X1, X2)) 50.42/13.82 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 50.42/13.82 x(mark(X1), X2) -> mark(x(X1, X2)) 50.42/13.82 x(X1, mark(X2)) -> mark(x(X1, X2)) 50.42/13.82 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 50.42/13.82 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 50.42/13.82 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 50.42/13.82 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 50.42/13.82 U15(mark(X1), X2) -> mark(U15(X1, X2)) 50.42/13.82 U16(mark(X)) -> mark(U16(X)) 50.42/13.82 U21(mark(X1), X2) -> mark(U21(X1, X2)) 50.42/13.82 U22(mark(X1), X2) -> mark(U22(X1, X2)) 50.42/13.82 U23(mark(X)) -> mark(U23(X)) 50.42/13.82 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 50.42/13.82 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 50.42/13.82 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 50.42/13.82 U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) 50.42/13.82 U35(mark(X1), X2) -> mark(U35(X1, X2)) 50.42/13.82 U36(mark(X)) -> mark(U36(X)) 50.42/13.82 U41(mark(X1), X2) -> mark(U41(X1, X2)) 50.42/13.82 U42(mark(X)) -> mark(U42(X)) 50.42/13.82 U51(mark(X)) -> mark(U51(X)) 50.42/13.82 U61(mark(X1), X2) -> mark(U61(X1, X2)) 50.42/13.82 U62(mark(X)) -> mark(U62(X)) 50.42/13.82 U71(mark(X1), X2) -> mark(U71(X1, X2)) 50.42/13.82 U72(mark(X1), X2) -> mark(U72(X1, X2)) 50.42/13.82 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 50.42/13.82 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 50.42/13.82 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 50.42/13.82 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 50.42/13.82 s(mark(X)) -> mark(s(X)) 50.42/13.82 U91(mark(X1), X2) -> mark(U91(X1, X2)) 50.42/13.82 U92(mark(X)) -> mark(U92(X)) 50.42/13.82 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(tt) -> ok(tt) 50.42/13.82 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(isNatKind(X)) -> isNatKind(proper(X)) 50.42/13.82 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(isNat(X)) -> isNat(proper(X)) 50.42/13.82 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 50.42/13.82 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 50.42/13.82 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) 50.42/13.82 proper(U16(X)) -> U16(proper(X)) 50.42/13.82 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 50.42/13.82 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 50.42/13.82 proper(U23(X)) -> U23(proper(X)) 50.42/13.82 proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) 50.42/13.82 proper(U36(X)) -> U36(proper(X)) 50.42/13.82 proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) 50.42/13.82 proper(U42(X)) -> U42(proper(X)) 50.42/13.82 proper(U51(X)) -> U51(proper(X)) 50.42/13.82 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 50.42/13.82 proper(U62(X)) -> U62(proper(X)) 50.42/13.82 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 50.42/13.82 proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) 50.42/13.82 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(s(X)) -> s(proper(X)) 50.42/13.82 proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) 50.42/13.82 proper(U92(X)) -> U92(proper(X)) 50.42/13.82 proper(0) -> ok(0) 50.42/13.82 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 50.42/13.82 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 50.42/13.82 isNatKind(ok(X)) -> ok(isNatKind(X)) 50.42/13.82 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 50.42/13.82 isNat(ok(X)) -> ok(isNat(X)) 50.42/13.82 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 50.42/13.82 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 50.42/13.82 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 50.42/13.82 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 50.42/13.82 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 50.42/13.82 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 50.42/13.82 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 50.42/13.82 U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) 50.42/13.82 U16(ok(X)) -> ok(U16(X)) 50.42/13.82 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 50.42/13.82 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 50.42/13.82 U23(ok(X)) -> ok(U23(X)) 50.42/13.82 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 50.42/13.82 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 50.42/13.82 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 50.42/13.82 U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) 50.42/13.82 U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) 50.42/13.82 U36(ok(X)) -> ok(U36(X)) 50.42/13.82 U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) 50.42/13.82 U42(ok(X)) -> ok(U42(X)) 50.42/13.82 U51(ok(X)) -> ok(U51(X)) 50.42/13.82 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 50.42/13.82 U62(ok(X)) -> ok(U62(X)) 50.42/13.82 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 50.42/13.82 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 50.42/13.82 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 50.42/13.82 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 50.42/13.82 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 50.42/13.82 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 50.42/13.82 s(ok(X)) -> ok(s(X)) 50.42/13.82 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 50.42/13.82 U92(ok(X)) -> ok(U92(X)) 50.42/13.82 top(mark(X)) -> top(proper(X)) 50.42/13.82 top(ok(X)) -> top(active(X)) 50.42/13.82 50.42/13.82 S is empty. 50.42/13.82 Rewrite Strategy: FULL 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (21) DecreasingLoopProof (LOWER BOUND(ID)) 50.42/13.82 The following loop(s) give(s) rise to the lower bound Omega(n^1): 50.42/13.82 50.42/13.82 The rewrite sequence 50.42/13.82 50.42/13.82 U23(mark(X)) ->^+ mark(U23(X)) 50.42/13.82 50.42/13.82 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 50.42/13.82 50.42/13.82 The pumping substitution is [X / mark(X)]. 50.42/13.82 50.42/13.82 The result substitution is [ ]. 50.42/13.82 50.42/13.82 50.42/13.82 50.42/13.82 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (22) 50.42/13.82 Complex Obligation (BEST) 50.42/13.82 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (23) 50.42/13.82 Obligation: 50.42/13.82 Proved the lower bound n^1 for the following obligation: 50.42/13.82 50.42/13.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 50.42/13.82 50.42/13.82 50.42/13.82 The TRS R consists of the following rules: 50.42/13.82 50.42/13.82 active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) 50.42/13.82 active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) 50.42/13.82 active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) 50.42/13.82 active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) 50.42/13.82 active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) 50.42/13.82 active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) 50.42/13.82 active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) 50.42/13.82 active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) 50.42/13.82 active(U15(tt, V2)) -> mark(U16(isNat(V2))) 50.42/13.82 active(U16(tt)) -> mark(tt) 50.42/13.82 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 50.42/13.82 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 50.42/13.82 active(U23(tt)) -> mark(tt) 50.42/13.82 active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) 50.42/13.82 active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) 50.42/13.82 active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) 50.42/13.82 active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) 50.42/13.82 active(U35(tt, V2)) -> mark(U36(isNat(V2))) 50.42/13.82 active(U36(tt)) -> mark(tt) 50.42/13.82 active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) 50.42/13.82 active(U42(tt)) -> mark(tt) 50.42/13.82 active(U51(tt)) -> mark(tt) 50.42/13.82 active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) 50.42/13.82 active(U62(tt)) -> mark(tt) 50.42/13.82 active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) 50.42/13.82 active(U72(tt, N)) -> mark(N) 50.42/13.82 active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) 50.42/13.82 active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) 50.42/13.82 active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) 50.42/13.82 active(U84(tt, M, N)) -> mark(s(plus(N, M))) 50.42/13.82 active(U91(tt, N)) -> mark(U92(isNatKind(N))) 50.42/13.82 active(U92(tt)) -> mark(0) 50.42/13.82 active(isNat(0)) -> mark(tt) 50.42/13.82 active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) 50.42/13.82 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 50.42/13.82 active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) 50.42/13.82 active(isNatKind(0)) -> mark(tt) 50.42/13.82 active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) 50.42/13.82 active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) 50.42/13.82 active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 50.42/13.82 active(plus(N, 0)) -> mark(U71(isNat(N), N)) 50.42/13.82 active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) 50.42/13.82 active(x(N, 0)) -> mark(U91(isNat(N), N)) 50.42/13.82 active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 50.42/13.82 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 50.42/13.82 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 50.42/13.82 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 50.42/13.82 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 50.42/13.82 active(plus(X1, X2)) -> plus(active(X1), X2) 50.42/13.82 active(plus(X1, X2)) -> plus(X1, active(X2)) 50.42/13.82 active(x(X1, X2)) -> x(active(X1), X2) 50.42/13.82 active(x(X1, X2)) -> x(X1, active(X2)) 50.42/13.82 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 50.42/13.82 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 50.42/13.82 active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) 50.42/13.82 active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) 50.42/13.82 active(U15(X1, X2)) -> U15(active(X1), X2) 50.42/13.82 active(U16(X)) -> U16(active(X)) 50.42/13.82 active(U21(X1, X2)) -> U21(active(X1), X2) 50.42/13.82 active(U22(X1, X2)) -> U22(active(X1), X2) 50.42/13.82 active(U23(X)) -> U23(active(X)) 50.42/13.82 active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) 50.42/13.82 active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) 50.42/13.82 active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) 50.42/13.82 active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) 50.42/13.82 active(U35(X1, X2)) -> U35(active(X1), X2) 50.42/13.82 active(U36(X)) -> U36(active(X)) 50.42/13.82 active(U41(X1, X2)) -> U41(active(X1), X2) 50.42/13.82 active(U42(X)) -> U42(active(X)) 50.42/13.82 active(U51(X)) -> U51(active(X)) 50.42/13.82 active(U61(X1, X2)) -> U61(active(X1), X2) 50.42/13.82 active(U62(X)) -> U62(active(X)) 50.42/13.82 active(U71(X1, X2)) -> U71(active(X1), X2) 50.42/13.82 active(U72(X1, X2)) -> U72(active(X1), X2) 50.42/13.82 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 50.42/13.82 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 50.42/13.82 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 50.42/13.82 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 50.42/13.82 active(s(X)) -> s(active(X)) 50.42/13.82 active(U91(X1, X2)) -> U91(active(X1), X2) 50.42/13.82 active(U92(X)) -> U92(active(X)) 50.42/13.82 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 50.42/13.82 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 50.42/13.82 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 50.42/13.82 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 50.42/13.82 plus(mark(X1), X2) -> mark(plus(X1, X2)) 50.42/13.82 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 50.42/13.82 x(mark(X1), X2) -> mark(x(X1, X2)) 50.42/13.82 x(X1, mark(X2)) -> mark(x(X1, X2)) 50.42/13.82 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 50.42/13.82 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 50.42/13.82 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 50.42/13.82 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 50.42/13.82 U15(mark(X1), X2) -> mark(U15(X1, X2)) 50.42/13.82 U16(mark(X)) -> mark(U16(X)) 50.42/13.82 U21(mark(X1), X2) -> mark(U21(X1, X2)) 50.42/13.82 U22(mark(X1), X2) -> mark(U22(X1, X2)) 50.42/13.82 U23(mark(X)) -> mark(U23(X)) 50.42/13.82 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 50.42/13.82 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 50.42/13.82 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 50.42/13.82 U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) 50.42/13.82 U35(mark(X1), X2) -> mark(U35(X1, X2)) 50.42/13.82 U36(mark(X)) -> mark(U36(X)) 50.42/13.82 U41(mark(X1), X2) -> mark(U41(X1, X2)) 50.42/13.82 U42(mark(X)) -> mark(U42(X)) 50.42/13.82 U51(mark(X)) -> mark(U51(X)) 50.42/13.82 U61(mark(X1), X2) -> mark(U61(X1, X2)) 50.42/13.82 U62(mark(X)) -> mark(U62(X)) 50.42/13.82 U71(mark(X1), X2) -> mark(U71(X1, X2)) 50.42/13.82 U72(mark(X1), X2) -> mark(U72(X1, X2)) 50.42/13.82 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 50.42/13.82 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 50.42/13.82 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 50.42/13.82 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 50.42/13.82 s(mark(X)) -> mark(s(X)) 50.42/13.82 U91(mark(X1), X2) -> mark(U91(X1, X2)) 50.42/13.82 U92(mark(X)) -> mark(U92(X)) 50.42/13.82 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(tt) -> ok(tt) 50.42/13.82 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(isNatKind(X)) -> isNatKind(proper(X)) 50.42/13.82 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(isNat(X)) -> isNat(proper(X)) 50.42/13.82 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 50.42/13.82 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 50.42/13.82 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) 50.42/13.82 proper(U16(X)) -> U16(proper(X)) 50.42/13.82 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 50.42/13.82 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 50.42/13.82 proper(U23(X)) -> U23(proper(X)) 50.42/13.82 proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) 50.42/13.82 proper(U36(X)) -> U36(proper(X)) 50.42/13.82 proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) 50.42/13.82 proper(U42(X)) -> U42(proper(X)) 50.42/13.82 proper(U51(X)) -> U51(proper(X)) 50.42/13.82 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 50.42/13.82 proper(U62(X)) -> U62(proper(X)) 50.42/13.82 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 50.42/13.82 proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) 50.42/13.82 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(s(X)) -> s(proper(X)) 50.42/13.82 proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) 50.42/13.82 proper(U92(X)) -> U92(proper(X)) 50.42/13.82 proper(0) -> ok(0) 50.42/13.82 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 50.42/13.82 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 50.42/13.82 isNatKind(ok(X)) -> ok(isNatKind(X)) 50.42/13.82 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 50.42/13.82 isNat(ok(X)) -> ok(isNat(X)) 50.42/13.82 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 50.42/13.82 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 50.42/13.82 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 50.42/13.82 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 50.42/13.82 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 50.42/13.82 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 50.42/13.82 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 50.42/13.82 U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) 50.42/13.82 U16(ok(X)) -> ok(U16(X)) 50.42/13.82 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 50.42/13.82 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 50.42/13.82 U23(ok(X)) -> ok(U23(X)) 50.42/13.82 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 50.42/13.82 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 50.42/13.82 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 50.42/13.82 U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) 50.42/13.82 U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) 50.42/13.82 U36(ok(X)) -> ok(U36(X)) 50.42/13.82 U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) 50.42/13.82 U42(ok(X)) -> ok(U42(X)) 50.42/13.82 U51(ok(X)) -> ok(U51(X)) 50.42/13.82 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 50.42/13.82 U62(ok(X)) -> ok(U62(X)) 50.42/13.82 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 50.42/13.82 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 50.42/13.82 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 50.42/13.82 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 50.42/13.82 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 50.42/13.82 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 50.42/13.82 s(ok(X)) -> ok(s(X)) 50.42/13.82 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 50.42/13.82 U92(ok(X)) -> ok(U92(X)) 50.42/13.82 top(mark(X)) -> top(proper(X)) 50.42/13.82 top(ok(X)) -> top(active(X)) 50.42/13.82 50.42/13.82 S is empty. 50.42/13.82 Rewrite Strategy: FULL 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (24) LowerBoundPropagationProof (FINISHED) 50.42/13.82 Propagated lower bound. 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (25) 50.42/13.82 BOUNDS(n^1, INF) 50.42/13.82 50.42/13.82 ---------------------------------------- 50.42/13.82 50.42/13.82 (26) 50.42/13.82 Obligation: 50.42/13.82 Analyzing the following TRS for decreasing loops: 50.42/13.82 50.42/13.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 50.42/13.82 50.42/13.82 50.42/13.82 The TRS R consists of the following rules: 50.42/13.82 50.42/13.82 active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) 50.42/13.82 active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) 50.42/13.82 active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) 50.42/13.82 active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) 50.42/13.82 active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) 50.42/13.82 active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) 50.42/13.82 active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) 50.42/13.82 active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) 50.42/13.82 active(U15(tt, V2)) -> mark(U16(isNat(V2))) 50.42/13.82 active(U16(tt)) -> mark(tt) 50.42/13.82 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 50.42/13.82 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 50.42/13.82 active(U23(tt)) -> mark(tt) 50.42/13.82 active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) 50.42/13.82 active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) 50.42/13.82 active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) 50.42/13.82 active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) 50.42/13.82 active(U35(tt, V2)) -> mark(U36(isNat(V2))) 50.42/13.82 active(U36(tt)) -> mark(tt) 50.42/13.82 active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) 50.42/13.82 active(U42(tt)) -> mark(tt) 50.42/13.82 active(U51(tt)) -> mark(tt) 50.42/13.82 active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) 50.42/13.82 active(U62(tt)) -> mark(tt) 50.42/13.82 active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) 50.42/13.82 active(U72(tt, N)) -> mark(N) 50.42/13.82 active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) 50.42/13.82 active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) 50.42/13.82 active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) 50.42/13.82 active(U84(tt, M, N)) -> mark(s(plus(N, M))) 50.42/13.82 active(U91(tt, N)) -> mark(U92(isNatKind(N))) 50.42/13.82 active(U92(tt)) -> mark(0) 50.42/13.82 active(isNat(0)) -> mark(tt) 50.42/13.82 active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) 50.42/13.82 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 50.42/13.82 active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) 50.42/13.82 active(isNatKind(0)) -> mark(tt) 50.42/13.82 active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) 50.42/13.82 active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) 50.42/13.82 active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 50.42/13.82 active(plus(N, 0)) -> mark(U71(isNat(N), N)) 50.42/13.82 active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) 50.42/13.82 active(x(N, 0)) -> mark(U91(isNat(N), N)) 50.42/13.82 active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) 50.42/13.82 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 50.42/13.82 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 50.42/13.82 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 50.42/13.82 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 50.42/13.82 active(plus(X1, X2)) -> plus(active(X1), X2) 50.42/13.82 active(plus(X1, X2)) -> plus(X1, active(X2)) 50.42/13.82 active(x(X1, X2)) -> x(active(X1), X2) 50.42/13.82 active(x(X1, X2)) -> x(X1, active(X2)) 50.42/13.82 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 50.42/13.82 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 50.42/13.82 active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) 50.42/13.82 active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) 50.42/13.82 active(U15(X1, X2)) -> U15(active(X1), X2) 50.42/13.82 active(U16(X)) -> U16(active(X)) 50.42/13.82 active(U21(X1, X2)) -> U21(active(X1), X2) 50.42/13.82 active(U22(X1, X2)) -> U22(active(X1), X2) 50.42/13.82 active(U23(X)) -> U23(active(X)) 50.42/13.82 active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) 50.42/13.82 active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) 50.42/13.82 active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) 50.42/13.82 active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) 50.42/13.82 active(U35(X1, X2)) -> U35(active(X1), X2) 50.42/13.82 active(U36(X)) -> U36(active(X)) 50.42/13.82 active(U41(X1, X2)) -> U41(active(X1), X2) 50.42/13.82 active(U42(X)) -> U42(active(X)) 50.42/13.82 active(U51(X)) -> U51(active(X)) 50.42/13.82 active(U61(X1, X2)) -> U61(active(X1), X2) 50.42/13.82 active(U62(X)) -> U62(active(X)) 50.42/13.82 active(U71(X1, X2)) -> U71(active(X1), X2) 50.42/13.82 active(U72(X1, X2)) -> U72(active(X1), X2) 50.42/13.82 active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) 50.42/13.82 active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) 50.42/13.82 active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) 50.42/13.82 active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) 50.42/13.82 active(s(X)) -> s(active(X)) 50.42/13.82 active(U91(X1, X2)) -> U91(active(X1), X2) 50.42/13.82 active(U92(X)) -> U92(active(X)) 50.42/13.82 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 50.42/13.82 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 50.42/13.82 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 50.42/13.82 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 50.42/13.82 plus(mark(X1), X2) -> mark(plus(X1, X2)) 50.42/13.82 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 50.42/13.82 x(mark(X1), X2) -> mark(x(X1, X2)) 50.42/13.82 x(X1, mark(X2)) -> mark(x(X1, X2)) 50.42/13.82 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 50.42/13.82 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 50.42/13.82 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 50.42/13.82 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 50.42/13.82 U15(mark(X1), X2) -> mark(U15(X1, X2)) 50.42/13.82 U16(mark(X)) -> mark(U16(X)) 50.42/13.82 U21(mark(X1), X2) -> mark(U21(X1, X2)) 50.42/13.82 U22(mark(X1), X2) -> mark(U22(X1, X2)) 50.42/13.82 U23(mark(X)) -> mark(U23(X)) 50.42/13.82 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 50.42/13.82 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 50.42/13.82 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 50.42/13.82 U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) 50.42/13.82 U35(mark(X1), X2) -> mark(U35(X1, X2)) 50.42/13.82 U36(mark(X)) -> mark(U36(X)) 50.42/13.82 U41(mark(X1), X2) -> mark(U41(X1, X2)) 50.42/13.82 U42(mark(X)) -> mark(U42(X)) 50.42/13.82 U51(mark(X)) -> mark(U51(X)) 50.42/13.82 U61(mark(X1), X2) -> mark(U61(X1, X2)) 50.42/13.82 U62(mark(X)) -> mark(U62(X)) 50.42/13.82 U71(mark(X1), X2) -> mark(U71(X1, X2)) 50.42/13.82 U72(mark(X1), X2) -> mark(U72(X1, X2)) 50.42/13.82 U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) 50.42/13.82 U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) 50.42/13.82 U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) 50.42/13.82 U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) 50.42/13.82 s(mark(X)) -> mark(s(X)) 50.42/13.82 U91(mark(X1), X2) -> mark(U91(X1, X2)) 50.42/13.82 U92(mark(X)) -> mark(U92(X)) 50.42/13.82 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(tt) -> ok(tt) 50.42/13.82 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(isNatKind(X)) -> isNatKind(proper(X)) 50.42/13.82 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(isNat(X)) -> isNat(proper(X)) 50.42/13.82 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 50.42/13.82 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 50.42/13.82 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) 50.42/13.82 proper(U16(X)) -> U16(proper(X)) 50.42/13.82 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 50.42/13.82 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 50.42/13.82 proper(U23(X)) -> U23(proper(X)) 50.42/13.82 proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) 50.42/13.82 proper(U36(X)) -> U36(proper(X)) 50.42/13.82 proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) 50.42/13.82 proper(U42(X)) -> U42(proper(X)) 50.42/13.82 proper(U51(X)) -> U51(proper(X)) 50.42/13.82 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 50.42/13.82 proper(U62(X)) -> U62(proper(X)) 50.42/13.82 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 50.42/13.82 proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) 50.42/13.82 proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) 50.42/13.82 proper(s(X)) -> s(proper(X)) 50.42/13.82 proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) 50.42/13.82 proper(U92(X)) -> U92(proper(X)) 50.42/13.82 proper(0) -> ok(0) 50.42/13.82 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 50.42/13.82 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 50.42/13.82 isNatKind(ok(X)) -> ok(isNatKind(X)) 50.42/13.82 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 50.42/13.82 isNat(ok(X)) -> ok(isNat(X)) 50.42/13.82 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 50.42/13.82 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 50.42/13.82 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 50.42/13.82 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 50.42/13.82 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 50.42/13.82 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 50.42/13.82 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 50.42/13.82 U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) 50.42/13.82 U16(ok(X)) -> ok(U16(X)) 50.42/13.82 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 50.42/13.82 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 50.42/13.82 U23(ok(X)) -> ok(U23(X)) 50.42/13.82 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 50.42/13.82 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 50.42/13.82 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 50.42/13.82 U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) 50.42/13.82 U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) 50.42/13.82 U36(ok(X)) -> ok(U36(X)) 50.42/13.82 U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) 50.42/13.82 U42(ok(X)) -> ok(U42(X)) 50.42/13.82 U51(ok(X)) -> ok(U51(X)) 50.42/13.82 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 50.42/13.82 U62(ok(X)) -> ok(U62(X)) 50.42/13.82 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 50.42/13.82 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 50.42/13.82 U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) 50.42/13.82 U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) 50.42/13.82 U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) 50.42/13.82 U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) 50.42/13.82 s(ok(X)) -> ok(s(X)) 50.42/13.82 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 50.42/13.82 U92(ok(X)) -> ok(U92(X)) 50.42/13.82 top(mark(X)) -> top(proper(X)) 50.42/13.82 top(ok(X)) -> top(active(X)) 50.42/13.82 50.42/13.82 S is empty. 50.42/13.82 Rewrite Strategy: FULL 50.45/13.86 EOF