WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 22.4 s] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (6) BEST (7) proven lower bound (8) LowerBoundPropagationProof [FINISHED, 0 ms] (9) BOUNDS(n^1, INF) (10) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U101(tt, z0, z1) -> c(A__U102(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U101(z0, z1, z2) -> c1 A__U102(tt, z0, z1) -> c2(A__U103(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U102(z0, z1, z2) -> c3 A__U103(tt, z0, z1) -> c4(A__U104(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U103(z0, z1, z2) -> c5 A__U104(tt, z0, z1) -> c6(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U104(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U104(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U104(z0, z1, z2) -> c9 A__U11(tt, z0, z1) -> c10(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c11 A__U12(tt, z0, z1) -> c12(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c13 A__U13(tt, z0, z1) -> c14(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c15 A__U14(tt, z0, z1) -> c16(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c17 A__U15(tt, z0) -> c18(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c19 A__U16(tt) -> c20 A__U16(z0) -> c21 A__U21(tt, z0) -> c22(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c23 A__U22(tt, z0) -> c24(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c25 A__U23(tt) -> c26 A__U23(z0) -> c27 A__U31(tt, z0, z1) -> c28(A__U32(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U31(z0, z1, z2) -> c29 A__U32(tt, z0, z1) -> c30(A__U33(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U32(z0, z1, z2) -> c31 A__U33(tt, z0, z1) -> c32(A__U34(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U33(z0, z1, z2) -> c33 A__U34(tt, z0, z1) -> c34(A__U35(a__isNat(z0), z1), A__ISNAT(z0)) A__U34(z0, z1, z2) -> c35 A__U35(tt, z0) -> c36(A__U36(a__isNat(z0)), A__ISNAT(z0)) A__U35(z0, z1) -> c37 A__U36(tt) -> c38 A__U36(z0) -> c39 A__U41(tt, z0) -> c40(A__U42(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U41(z0, z1) -> c41 A__U42(tt) -> c42 A__U42(z0) -> c43 A__U51(tt) -> c44 A__U51(z0) -> c45 A__U61(tt, z0) -> c46(A__U62(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U61(z0, z1) -> c47 A__U62(tt) -> c48 A__U62(z0) -> c49 A__U71(tt, z0) -> c50(A__U72(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U71(z0, z1) -> c51 A__U72(tt, z0) -> c52(MARK(z0)) A__U72(z0, z1) -> c53 A__U81(tt, z0, z1) -> c54(A__U82(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U81(z0, z1, z2) -> c55 A__U82(tt, z0, z1) -> c56(A__U83(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U82(z0, z1, z2) -> c57 A__U83(tt, z0, z1) -> c58(A__U84(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U83(z0, z1, z2) -> c59 A__U84(tt, z0, z1) -> c60(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U84(tt, z0, z1) -> c61(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U84(z0, z1, z2) -> c62 A__U91(tt, z0) -> c63(A__U92(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U91(z0, z1) -> c64 A__U92(tt) -> c65 A__U92(z0) -> c66 A__ISNAT(0) -> c67 A__ISNAT(plus(z0, z1)) -> c68(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c69(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(x(z0, z1)) -> c70(A__U31(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(z0) -> c71 A__ISNATKIND(0) -> c72 A__ISNATKIND(plus(z0, z1)) -> c73(A__U41(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c74(A__U51(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(x(z0, z1)) -> c75(A__U61(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c76 A__PLUS(z0, 0) -> c77(A__U71(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c78(A__U81(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c79 A__X(z0, 0) -> c80(A__U91(a__isNat(z0), z0), A__ISNAT(z0)) A__X(z0, s(z1)) -> c81(A__U101(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c82 MARK(U101(z0, z1, z2)) -> c83(A__U101(mark(z0), z1, z2), MARK(z0)) MARK(U102(z0, z1, z2)) -> c84(A__U102(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c85(A__ISNATKIND(z0)) MARK(U103(z0, z1, z2)) -> c86(A__U103(mark(z0), z1, z2), MARK(z0)) MARK(isNat(z0)) -> c87(A__ISNAT(z0)) MARK(U104(z0, z1, z2)) -> c88(A__U104(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c89(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c90(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(x(z0, z1)) -> c91(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c92(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(U11(z0, z1, z2)) -> c93(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c94(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(U13(z0, z1, z2)) -> c95(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c96(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c97(A__U15(mark(z0), z1), MARK(z0)) MARK(U16(z0)) -> c98(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c99(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c100(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c101(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1, z2)) -> c102(A__U31(mark(z0), z1, z2), MARK(z0)) MARK(U32(z0, z1, z2)) -> c103(A__U32(mark(z0), z1, z2), MARK(z0)) MARK(U33(z0, z1, z2)) -> c104(A__U33(mark(z0), z1, z2), MARK(z0)) MARK(U34(z0, z1, z2)) -> c105(A__U34(mark(z0), z1, z2), MARK(z0)) MARK(U35(z0, z1)) -> c106(A__U35(mark(z0), z1), MARK(z0)) MARK(U36(z0)) -> c107(A__U36(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c108(A__U41(mark(z0), z1), MARK(z0)) MARK(U42(z0)) -> c109(A__U42(mark(z0)), MARK(z0)) MARK(U51(z0)) -> c110(A__U51(mark(z0)), MARK(z0)) MARK(U61(z0, z1)) -> c111(A__U61(mark(z0), z1), MARK(z0)) MARK(U62(z0)) -> c112(A__U62(mark(z0)), MARK(z0)) MARK(U71(z0, z1)) -> c113(A__U71(mark(z0), z1), MARK(z0)) MARK(U72(z0, z1)) -> c114(A__U72(mark(z0), z1), MARK(z0)) MARK(U81(z0, z1, z2)) -> c115(A__U81(mark(z0), z1, z2), MARK(z0)) MARK(U82(z0, z1, z2)) -> c116(A__U82(mark(z0), z1, z2), MARK(z0)) MARK(U83(z0, z1, z2)) -> c117(A__U83(mark(z0), z1, z2), MARK(z0)) MARK(U84(z0, z1, z2)) -> c118(A__U84(mark(z0), z1, z2), MARK(z0)) MARK(U91(z0, z1)) -> c119(A__U91(mark(z0), z1), MARK(z0)) MARK(U92(z0)) -> c120(A__U92(mark(z0)), MARK(z0)) MARK(tt) -> c121 MARK(s(z0)) -> c122(MARK(z0)) MARK(0) -> c123 The (relative) TRS S consists of the following rules: a__U101(tt, z0, z1) -> a__U102(a__isNatKind(z0), z0, z1) a__U101(z0, z1, z2) -> U101(z0, z1, z2) a__U102(tt, z0, z1) -> a__U103(a__isNat(z1), z0, z1) a__U102(z0, z1, z2) -> U102(z0, z1, z2) a__U103(tt, z0, z1) -> a__U104(a__isNatKind(z1), z0, z1) a__U103(z0, z1, z2) -> U103(z0, z1, z2) a__U104(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U104(z0, z1, z2) -> U104(z0, z1, z2) a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0, z1) -> a__U32(a__isNatKind(z0), z0, z1) a__U31(z0, z1, z2) -> U31(z0, z1, z2) a__U32(tt, z0, z1) -> a__U33(a__isNatKind(z1), z0, z1) a__U32(z0, z1, z2) -> U32(z0, z1, z2) a__U33(tt, z0, z1) -> a__U34(a__isNatKind(z1), z0, z1) a__U33(z0, z1, z2) -> U33(z0, z1, z2) a__U34(tt, z0, z1) -> a__U35(a__isNat(z0), z1) a__U34(z0, z1, z2) -> U34(z0, z1, z2) a__U35(tt, z0) -> a__U36(a__isNat(z0)) a__U35(z0, z1) -> U35(z0, z1) a__U36(tt) -> tt a__U36(z0) -> U36(z0) a__U41(tt, z0) -> a__U42(a__isNatKind(z0)) a__U41(z0, z1) -> U41(z0, z1) a__U42(tt) -> tt a__U42(z0) -> U42(z0) a__U51(tt) -> tt a__U51(z0) -> U51(z0) a__U61(tt, z0) -> a__U62(a__isNatKind(z0)) a__U61(z0, z1) -> U61(z0, z1) a__U62(tt) -> tt a__U62(z0) -> U62(z0) a__U71(tt, z0) -> a__U72(a__isNatKind(z0), z0) a__U71(z0, z1) -> U71(z0, z1) a__U72(tt, z0) -> mark(z0) a__U72(z0, z1) -> U72(z0, z1) a__U81(tt, z0, z1) -> a__U82(a__isNatKind(z0), z0, z1) a__U81(z0, z1, z2) -> U81(z0, z1, z2) a__U82(tt, z0, z1) -> a__U83(a__isNat(z1), z0, z1) a__U82(z0, z1, z2) -> U82(z0, z1, z2) a__U83(tt, z0, z1) -> a__U84(a__isNatKind(z1), z0, z1) a__U83(z0, z1, z2) -> U83(z0, z1, z2) a__U84(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U84(z0, z1, z2) -> U84(z0, z1, z2) a__U91(tt, z0) -> a__U92(a__isNatKind(z0)) a__U91(z0, z1) -> U91(z0, z1) a__U92(tt) -> 0 a__U92(z0) -> U92(z0) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(x(z0, z1)) -> a__U31(a__isNatKind(z0), z0, z1) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U41(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U51(a__isNatKind(z0)) a__isNatKind(x(z0, z1)) -> a__U61(a__isNatKind(z0), z1) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U71(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U81(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U91(a__isNat(z0), z0) a__x(z0, s(z1)) -> a__U101(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U101(z0, z1, z2)) -> a__U101(mark(z0), z1, z2) mark(U102(z0, z1, z2)) -> a__U102(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U103(z0, z1, z2)) -> a__U103(mark(z0), z1, z2) mark(isNat(z0)) -> a__isNat(z0) mark(U104(z0, z1, z2)) -> a__U104(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1, z2)) -> a__U31(mark(z0), z1, z2) mark(U32(z0, z1, z2)) -> a__U32(mark(z0), z1, z2) mark(U33(z0, z1, z2)) -> a__U33(mark(z0), z1, z2) mark(U34(z0, z1, z2)) -> a__U34(mark(z0), z1, z2) mark(U35(z0, z1)) -> a__U35(mark(z0), z1) mark(U36(z0)) -> a__U36(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U42(z0)) -> a__U42(mark(z0)) mark(U51(z0)) -> a__U51(mark(z0)) mark(U61(z0, z1)) -> a__U61(mark(z0), z1) mark(U62(z0)) -> a__U62(mark(z0)) mark(U71(z0, z1)) -> a__U71(mark(z0), z1) mark(U72(z0, z1)) -> a__U72(mark(z0), z1) mark(U81(z0, z1, z2)) -> a__U81(mark(z0), z1, z2) mark(U82(z0, z1, z2)) -> a__U82(mark(z0), z1, z2) mark(U83(z0, z1, z2)) -> a__U83(mark(z0), z1, z2) mark(U84(z0, z1, z2)) -> a__U84(mark(z0), z1, z2) mark(U91(z0, z1)) -> a__U91(mark(z0), z1) mark(U92(z0)) -> a__U92(mark(z0)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U101(tt, z0, z1) -> c(A__U102(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U101(z0, z1, z2) -> c1 A__U102(tt, z0, z1) -> c2(A__U103(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U102(z0, z1, z2) -> c3 A__U103(tt, z0, z1) -> c4(A__U104(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U103(z0, z1, z2) -> c5 A__U104(tt, z0, z1) -> c6(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U104(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U104(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U104(z0, z1, z2) -> c9 A__U11(tt, z0, z1) -> c10(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c11 A__U12(tt, z0, z1) -> c12(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c13 A__U13(tt, z0, z1) -> c14(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c15 A__U14(tt, z0, z1) -> c16(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c17 A__U15(tt, z0) -> c18(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c19 A__U16(tt) -> c20 A__U16(z0) -> c21 A__U21(tt, z0) -> c22(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c23 A__U22(tt, z0) -> c24(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c25 A__U23(tt) -> c26 A__U23(z0) -> c27 A__U31(tt, z0, z1) -> c28(A__U32(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U31(z0, z1, z2) -> c29 A__U32(tt, z0, z1) -> c30(A__U33(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U32(z0, z1, z2) -> c31 A__U33(tt, z0, z1) -> c32(A__U34(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U33(z0, z1, z2) -> c33 A__U34(tt, z0, z1) -> c34(A__U35(a__isNat(z0), z1), A__ISNAT(z0)) A__U34(z0, z1, z2) -> c35 A__U35(tt, z0) -> c36(A__U36(a__isNat(z0)), A__ISNAT(z0)) A__U35(z0, z1) -> c37 A__U36(tt) -> c38 A__U36(z0) -> c39 A__U41(tt, z0) -> c40(A__U42(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U41(z0, z1) -> c41 A__U42(tt) -> c42 A__U42(z0) -> c43 A__U51(tt) -> c44 A__U51(z0) -> c45 A__U61(tt, z0) -> c46(A__U62(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U61(z0, z1) -> c47 A__U62(tt) -> c48 A__U62(z0) -> c49 A__U71(tt, z0) -> c50(A__U72(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U71(z0, z1) -> c51 A__U72(tt, z0) -> c52(MARK(z0)) A__U72(z0, z1) -> c53 A__U81(tt, z0, z1) -> c54(A__U82(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U81(z0, z1, z2) -> c55 A__U82(tt, z0, z1) -> c56(A__U83(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U82(z0, z1, z2) -> c57 A__U83(tt, z0, z1) -> c58(A__U84(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U83(z0, z1, z2) -> c59 A__U84(tt, z0, z1) -> c60(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U84(tt, z0, z1) -> c61(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U84(z0, z1, z2) -> c62 A__U91(tt, z0) -> c63(A__U92(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U91(z0, z1) -> c64 A__U92(tt) -> c65 A__U92(z0) -> c66 A__ISNAT(0) -> c67 A__ISNAT(plus(z0, z1)) -> c68(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c69(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(x(z0, z1)) -> c70(A__U31(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(z0) -> c71 A__ISNATKIND(0) -> c72 A__ISNATKIND(plus(z0, z1)) -> c73(A__U41(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c74(A__U51(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(x(z0, z1)) -> c75(A__U61(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c76 A__PLUS(z0, 0) -> c77(A__U71(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c78(A__U81(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c79 A__X(z0, 0) -> c80(A__U91(a__isNat(z0), z0), A__ISNAT(z0)) A__X(z0, s(z1)) -> c81(A__U101(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c82 MARK(U101(z0, z1, z2)) -> c83(A__U101(mark(z0), z1, z2), MARK(z0)) MARK(U102(z0, z1, z2)) -> c84(A__U102(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c85(A__ISNATKIND(z0)) MARK(U103(z0, z1, z2)) -> c86(A__U103(mark(z0), z1, z2), MARK(z0)) MARK(isNat(z0)) -> c87(A__ISNAT(z0)) MARK(U104(z0, z1, z2)) -> c88(A__U104(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c89(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c90(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(x(z0, z1)) -> c91(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c92(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(U11(z0, z1, z2)) -> c93(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c94(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(U13(z0, z1, z2)) -> c95(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c96(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c97(A__U15(mark(z0), z1), MARK(z0)) MARK(U16(z0)) -> c98(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c99(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c100(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c101(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1, z2)) -> c102(A__U31(mark(z0), z1, z2), MARK(z0)) MARK(U32(z0, z1, z2)) -> c103(A__U32(mark(z0), z1, z2), MARK(z0)) MARK(U33(z0, z1, z2)) -> c104(A__U33(mark(z0), z1, z2), MARK(z0)) MARK(U34(z0, z1, z2)) -> c105(A__U34(mark(z0), z1, z2), MARK(z0)) MARK(U35(z0, z1)) -> c106(A__U35(mark(z0), z1), MARK(z0)) MARK(U36(z0)) -> c107(A__U36(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c108(A__U41(mark(z0), z1), MARK(z0)) MARK(U42(z0)) -> c109(A__U42(mark(z0)), MARK(z0)) MARK(U51(z0)) -> c110(A__U51(mark(z0)), MARK(z0)) MARK(U61(z0, z1)) -> c111(A__U61(mark(z0), z1), MARK(z0)) MARK(U62(z0)) -> c112(A__U62(mark(z0)), MARK(z0)) MARK(U71(z0, z1)) -> c113(A__U71(mark(z0), z1), MARK(z0)) MARK(U72(z0, z1)) -> c114(A__U72(mark(z0), z1), MARK(z0)) MARK(U81(z0, z1, z2)) -> c115(A__U81(mark(z0), z1, z2), MARK(z0)) MARK(U82(z0, z1, z2)) -> c116(A__U82(mark(z0), z1, z2), MARK(z0)) MARK(U83(z0, z1, z2)) -> c117(A__U83(mark(z0), z1, z2), MARK(z0)) MARK(U84(z0, z1, z2)) -> c118(A__U84(mark(z0), z1, z2), MARK(z0)) MARK(U91(z0, z1)) -> c119(A__U91(mark(z0), z1), MARK(z0)) MARK(U92(z0)) -> c120(A__U92(mark(z0)), MARK(z0)) MARK(tt) -> c121 MARK(s(z0)) -> c122(MARK(z0)) MARK(0) -> c123 The (relative) TRS S consists of the following rules: a__U101(tt, z0, z1) -> a__U102(a__isNatKind(z0), z0, z1) a__U101(z0, z1, z2) -> U101(z0, z1, z2) a__U102(tt, z0, z1) -> a__U103(a__isNat(z1), z0, z1) a__U102(z0, z1, z2) -> U102(z0, z1, z2) a__U103(tt, z0, z1) -> a__U104(a__isNatKind(z1), z0, z1) a__U103(z0, z1, z2) -> U103(z0, z1, z2) a__U104(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U104(z0, z1, z2) -> U104(z0, z1, z2) a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0, z1) -> a__U32(a__isNatKind(z0), z0, z1) a__U31(z0, z1, z2) -> U31(z0, z1, z2) a__U32(tt, z0, z1) -> a__U33(a__isNatKind(z1), z0, z1) a__U32(z0, z1, z2) -> U32(z0, z1, z2) a__U33(tt, z0, z1) -> a__U34(a__isNatKind(z1), z0, z1) a__U33(z0, z1, z2) -> U33(z0, z1, z2) a__U34(tt, z0, z1) -> a__U35(a__isNat(z0), z1) a__U34(z0, z1, z2) -> U34(z0, z1, z2) a__U35(tt, z0) -> a__U36(a__isNat(z0)) a__U35(z0, z1) -> U35(z0, z1) a__U36(tt) -> tt a__U36(z0) -> U36(z0) a__U41(tt, z0) -> a__U42(a__isNatKind(z0)) a__U41(z0, z1) -> U41(z0, z1) a__U42(tt) -> tt a__U42(z0) -> U42(z0) a__U51(tt) -> tt a__U51(z0) -> U51(z0) a__U61(tt, z0) -> a__U62(a__isNatKind(z0)) a__U61(z0, z1) -> U61(z0, z1) a__U62(tt) -> tt a__U62(z0) -> U62(z0) a__U71(tt, z0) -> a__U72(a__isNatKind(z0), z0) a__U71(z0, z1) -> U71(z0, z1) a__U72(tt, z0) -> mark(z0) a__U72(z0, z1) -> U72(z0, z1) a__U81(tt, z0, z1) -> a__U82(a__isNatKind(z0), z0, z1) a__U81(z0, z1, z2) -> U81(z0, z1, z2) a__U82(tt, z0, z1) -> a__U83(a__isNat(z1), z0, z1) a__U82(z0, z1, z2) -> U82(z0, z1, z2) a__U83(tt, z0, z1) -> a__U84(a__isNatKind(z1), z0, z1) a__U83(z0, z1, z2) -> U83(z0, z1, z2) a__U84(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U84(z0, z1, z2) -> U84(z0, z1, z2) a__U91(tt, z0) -> a__U92(a__isNatKind(z0)) a__U91(z0, z1) -> U91(z0, z1) a__U92(tt) -> 0 a__U92(z0) -> U92(z0) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(x(z0, z1)) -> a__U31(a__isNatKind(z0), z0, z1) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U41(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U51(a__isNatKind(z0)) a__isNatKind(x(z0, z1)) -> a__U61(a__isNatKind(z0), z1) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U71(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U81(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U91(a__isNat(z0), z0) a__x(z0, s(z1)) -> a__U101(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U101(z0, z1, z2)) -> a__U101(mark(z0), z1, z2) mark(U102(z0, z1, z2)) -> a__U102(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U103(z0, z1, z2)) -> a__U103(mark(z0), z1, z2) mark(isNat(z0)) -> a__isNat(z0) mark(U104(z0, z1, z2)) -> a__U104(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1, z2)) -> a__U31(mark(z0), z1, z2) mark(U32(z0, z1, z2)) -> a__U32(mark(z0), z1, z2) mark(U33(z0, z1, z2)) -> a__U33(mark(z0), z1, z2) mark(U34(z0, z1, z2)) -> a__U34(mark(z0), z1, z2) mark(U35(z0, z1)) -> a__U35(mark(z0), z1) mark(U36(z0)) -> a__U36(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U42(z0)) -> a__U42(mark(z0)) mark(U51(z0)) -> a__U51(mark(z0)) mark(U61(z0, z1)) -> a__U61(mark(z0), z1) mark(U62(z0)) -> a__U62(mark(z0)) mark(U71(z0, z1)) -> a__U71(mark(z0), z1) mark(U72(z0, z1)) -> a__U72(mark(z0), z1) mark(U81(z0, z1, z2)) -> a__U81(mark(z0), z1, z2) mark(U82(z0, z1, z2)) -> a__U82(mark(z0), z1, z2) mark(U83(z0, z1, z2)) -> a__U83(mark(z0), z1, z2) mark(U84(z0, z1, z2)) -> a__U84(mark(z0), z1, z2) mark(U91(z0, z1)) -> a__U91(mark(z0), z1) mark(U92(z0)) -> a__U92(mark(z0)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (4) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U101(tt, z0, z1) -> c(A__U102(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U101(z0, z1, z2) -> c1 A__U102(tt, z0, z1) -> c2(A__U103(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U102(z0, z1, z2) -> c3 A__U103(tt, z0, z1) -> c4(A__U104(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U103(z0, z1, z2) -> c5 A__U104(tt, z0, z1) -> c6(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U104(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U104(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U104(z0, z1, z2) -> c9 A__U11(tt, z0, z1) -> c10(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c11 A__U12(tt, z0, z1) -> c12(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c13 A__U13(tt, z0, z1) -> c14(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c15 A__U14(tt, z0, z1) -> c16(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c17 A__U15(tt, z0) -> c18(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c19 A__U16(tt) -> c20 A__U16(z0) -> c21 A__U21(tt, z0) -> c22(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c23 A__U22(tt, z0) -> c24(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c25 A__U23(tt) -> c26 A__U23(z0) -> c27 A__U31(tt, z0, z1) -> c28(A__U32(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U31(z0, z1, z2) -> c29 A__U32(tt, z0, z1) -> c30(A__U33(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U32(z0, z1, z2) -> c31 A__U33(tt, z0, z1) -> c32(A__U34(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U33(z0, z1, z2) -> c33 A__U34(tt, z0, z1) -> c34(A__U35(a__isNat(z0), z1), A__ISNAT(z0)) A__U34(z0, z1, z2) -> c35 A__U35(tt, z0) -> c36(A__U36(a__isNat(z0)), A__ISNAT(z0)) A__U35(z0, z1) -> c37 A__U36(tt) -> c38 A__U36(z0) -> c39 A__U41(tt, z0) -> c40(A__U42(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U41(z0, z1) -> c41 A__U42(tt) -> c42 A__U42(z0) -> c43 A__U51(tt) -> c44 A__U51(z0) -> c45 A__U61(tt, z0) -> c46(A__U62(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U61(z0, z1) -> c47 A__U62(tt) -> c48 A__U62(z0) -> c49 A__U71(tt, z0) -> c50(A__U72(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U71(z0, z1) -> c51 A__U72(tt, z0) -> c52(MARK(z0)) A__U72(z0, z1) -> c53 A__U81(tt, z0, z1) -> c54(A__U82(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U81(z0, z1, z2) -> c55 A__U82(tt, z0, z1) -> c56(A__U83(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U82(z0, z1, z2) -> c57 A__U83(tt, z0, z1) -> c58(A__U84(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U83(z0, z1, z2) -> c59 A__U84(tt, z0, z1) -> c60(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U84(tt, z0, z1) -> c61(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U84(z0, z1, z2) -> c62 A__U91(tt, z0) -> c63(A__U92(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U91(z0, z1) -> c64 A__U92(tt) -> c65 A__U92(z0) -> c66 A__ISNAT(0) -> c67 A__ISNAT(plus(z0, z1)) -> c68(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c69(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(x(z0, z1)) -> c70(A__U31(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(z0) -> c71 A__ISNATKIND(0) -> c72 A__ISNATKIND(plus(z0, z1)) -> c73(A__U41(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c74(A__U51(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(x(z0, z1)) -> c75(A__U61(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c76 A__PLUS(z0, 0) -> c77(A__U71(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c78(A__U81(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c79 A__X(z0, 0) -> c80(A__U91(a__isNat(z0), z0), A__ISNAT(z0)) A__X(z0, s(z1)) -> c81(A__U101(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c82 MARK(U101(z0, z1, z2)) -> c83(A__U101(mark(z0), z1, z2), MARK(z0)) MARK(U102(z0, z1, z2)) -> c84(A__U102(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c85(A__ISNATKIND(z0)) MARK(U103(z0, z1, z2)) -> c86(A__U103(mark(z0), z1, z2), MARK(z0)) MARK(isNat(z0)) -> c87(A__ISNAT(z0)) MARK(U104(z0, z1, z2)) -> c88(A__U104(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c89(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c90(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(x(z0, z1)) -> c91(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c92(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(U11(z0, z1, z2)) -> c93(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c94(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(U13(z0, z1, z2)) -> c95(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c96(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c97(A__U15(mark(z0), z1), MARK(z0)) MARK(U16(z0)) -> c98(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c99(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c100(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c101(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1, z2)) -> c102(A__U31(mark(z0), z1, z2), MARK(z0)) MARK(U32(z0, z1, z2)) -> c103(A__U32(mark(z0), z1, z2), MARK(z0)) MARK(U33(z0, z1, z2)) -> c104(A__U33(mark(z0), z1, z2), MARK(z0)) MARK(U34(z0, z1, z2)) -> c105(A__U34(mark(z0), z1, z2), MARK(z0)) MARK(U35(z0, z1)) -> c106(A__U35(mark(z0), z1), MARK(z0)) MARK(U36(z0)) -> c107(A__U36(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c108(A__U41(mark(z0), z1), MARK(z0)) MARK(U42(z0)) -> c109(A__U42(mark(z0)), MARK(z0)) MARK(U51(z0)) -> c110(A__U51(mark(z0)), MARK(z0)) MARK(U61(z0, z1)) -> c111(A__U61(mark(z0), z1), MARK(z0)) MARK(U62(z0)) -> c112(A__U62(mark(z0)), MARK(z0)) MARK(U71(z0, z1)) -> c113(A__U71(mark(z0), z1), MARK(z0)) MARK(U72(z0, z1)) -> c114(A__U72(mark(z0), z1), MARK(z0)) MARK(U81(z0, z1, z2)) -> c115(A__U81(mark(z0), z1, z2), MARK(z0)) MARK(U82(z0, z1, z2)) -> c116(A__U82(mark(z0), z1, z2), MARK(z0)) MARK(U83(z0, z1, z2)) -> c117(A__U83(mark(z0), z1, z2), MARK(z0)) MARK(U84(z0, z1, z2)) -> c118(A__U84(mark(z0), z1, z2), MARK(z0)) MARK(U91(z0, z1)) -> c119(A__U91(mark(z0), z1), MARK(z0)) MARK(U92(z0)) -> c120(A__U92(mark(z0)), MARK(z0)) MARK(tt) -> c121 MARK(s(z0)) -> c122(MARK(z0)) MARK(0) -> c123 The (relative) TRS S consists of the following rules: a__U101(tt, z0, z1) -> a__U102(a__isNatKind(z0), z0, z1) a__U101(z0, z1, z2) -> U101(z0, z1, z2) a__U102(tt, z0, z1) -> a__U103(a__isNat(z1), z0, z1) a__U102(z0, z1, z2) -> U102(z0, z1, z2) a__U103(tt, z0, z1) -> a__U104(a__isNatKind(z1), z0, z1) a__U103(z0, z1, z2) -> U103(z0, z1, z2) a__U104(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U104(z0, z1, z2) -> U104(z0, z1, z2) a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0, z1) -> a__U32(a__isNatKind(z0), z0, z1) a__U31(z0, z1, z2) -> U31(z0, z1, z2) a__U32(tt, z0, z1) -> a__U33(a__isNatKind(z1), z0, z1) a__U32(z0, z1, z2) -> U32(z0, z1, z2) a__U33(tt, z0, z1) -> a__U34(a__isNatKind(z1), z0, z1) a__U33(z0, z1, z2) -> U33(z0, z1, z2) a__U34(tt, z0, z1) -> a__U35(a__isNat(z0), z1) a__U34(z0, z1, z2) -> U34(z0, z1, z2) a__U35(tt, z0) -> a__U36(a__isNat(z0)) a__U35(z0, z1) -> U35(z0, z1) a__U36(tt) -> tt a__U36(z0) -> U36(z0) a__U41(tt, z0) -> a__U42(a__isNatKind(z0)) a__U41(z0, z1) -> U41(z0, z1) a__U42(tt) -> tt a__U42(z0) -> U42(z0) a__U51(tt) -> tt a__U51(z0) -> U51(z0) a__U61(tt, z0) -> a__U62(a__isNatKind(z0)) a__U61(z0, z1) -> U61(z0, z1) a__U62(tt) -> tt a__U62(z0) -> U62(z0) a__U71(tt, z0) -> a__U72(a__isNatKind(z0), z0) a__U71(z0, z1) -> U71(z0, z1) a__U72(tt, z0) -> mark(z0) a__U72(z0, z1) -> U72(z0, z1) a__U81(tt, z0, z1) -> a__U82(a__isNatKind(z0), z0, z1) a__U81(z0, z1, z2) -> U81(z0, z1, z2) a__U82(tt, z0, z1) -> a__U83(a__isNat(z1), z0, z1) a__U82(z0, z1, z2) -> U82(z0, z1, z2) a__U83(tt, z0, z1) -> a__U84(a__isNatKind(z1), z0, z1) a__U83(z0, z1, z2) -> U83(z0, z1, z2) a__U84(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U84(z0, z1, z2) -> U84(z0, z1, z2) a__U91(tt, z0) -> a__U92(a__isNatKind(z0)) a__U91(z0, z1) -> U91(z0, z1) a__U92(tt) -> 0 a__U92(z0) -> U92(z0) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(x(z0, z1)) -> a__U31(a__isNatKind(z0), z0, z1) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U41(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U51(a__isNatKind(z0)) a__isNatKind(x(z0, z1)) -> a__U61(a__isNatKind(z0), z1) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U71(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U81(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U91(a__isNat(z0), z0) a__x(z0, s(z1)) -> a__U101(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U101(z0, z1, z2)) -> a__U101(mark(z0), z1, z2) mark(U102(z0, z1, z2)) -> a__U102(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U103(z0, z1, z2)) -> a__U103(mark(z0), z1, z2) mark(isNat(z0)) -> a__isNat(z0) mark(U104(z0, z1, z2)) -> a__U104(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1, z2)) -> a__U31(mark(z0), z1, z2) mark(U32(z0, z1, z2)) -> a__U32(mark(z0), z1, z2) mark(U33(z0, z1, z2)) -> a__U33(mark(z0), z1, z2) mark(U34(z0, z1, z2)) -> a__U34(mark(z0), z1, z2) mark(U35(z0, z1)) -> a__U35(mark(z0), z1) mark(U36(z0)) -> a__U36(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U42(z0)) -> a__U42(mark(z0)) mark(U51(z0)) -> a__U51(mark(z0)) mark(U61(z0, z1)) -> a__U61(mark(z0), z1) mark(U62(z0)) -> a__U62(mark(z0)) mark(U71(z0, z1)) -> a__U71(mark(z0), z1) mark(U72(z0, z1)) -> a__U72(mark(z0), z1) mark(U81(z0, z1, z2)) -> a__U81(mark(z0), z1, z2) mark(U82(z0, z1, z2)) -> a__U82(mark(z0), z1, z2) mark(U83(z0, z1, z2)) -> a__U83(mark(z0), z1, z2) mark(U84(z0, z1, z2)) -> a__U84(mark(z0), z1, z2) mark(U91(z0, z1)) -> a__U91(mark(z0), z1) mark(U92(z0)) -> a__U92(mark(z0)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (5) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence MARK(U16(z0)) ->^+ c98(A__U16(mark(z0)), MARK(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / U16(z0)]. The result substitution is [ ]. ---------------------------------------- (6) Complex Obligation (BEST) ---------------------------------------- (7) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U101(tt, z0, z1) -> c(A__U102(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U101(z0, z1, z2) -> c1 A__U102(tt, z0, z1) -> c2(A__U103(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U102(z0, z1, z2) -> c3 A__U103(tt, z0, z1) -> c4(A__U104(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U103(z0, z1, z2) -> c5 A__U104(tt, z0, z1) -> c6(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U104(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U104(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U104(z0, z1, z2) -> c9 A__U11(tt, z0, z1) -> c10(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c11 A__U12(tt, z0, z1) -> c12(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c13 A__U13(tt, z0, z1) -> c14(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c15 A__U14(tt, z0, z1) -> c16(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c17 A__U15(tt, z0) -> c18(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c19 A__U16(tt) -> c20 A__U16(z0) -> c21 A__U21(tt, z0) -> c22(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c23 A__U22(tt, z0) -> c24(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c25 A__U23(tt) -> c26 A__U23(z0) -> c27 A__U31(tt, z0, z1) -> c28(A__U32(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U31(z0, z1, z2) -> c29 A__U32(tt, z0, z1) -> c30(A__U33(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U32(z0, z1, z2) -> c31 A__U33(tt, z0, z1) -> c32(A__U34(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U33(z0, z1, z2) -> c33 A__U34(tt, z0, z1) -> c34(A__U35(a__isNat(z0), z1), A__ISNAT(z0)) A__U34(z0, z1, z2) -> c35 A__U35(tt, z0) -> c36(A__U36(a__isNat(z0)), A__ISNAT(z0)) A__U35(z0, z1) -> c37 A__U36(tt) -> c38 A__U36(z0) -> c39 A__U41(tt, z0) -> c40(A__U42(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U41(z0, z1) -> c41 A__U42(tt) -> c42 A__U42(z0) -> c43 A__U51(tt) -> c44 A__U51(z0) -> c45 A__U61(tt, z0) -> c46(A__U62(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U61(z0, z1) -> c47 A__U62(tt) -> c48 A__U62(z0) -> c49 A__U71(tt, z0) -> c50(A__U72(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U71(z0, z1) -> c51 A__U72(tt, z0) -> c52(MARK(z0)) A__U72(z0, z1) -> c53 A__U81(tt, z0, z1) -> c54(A__U82(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U81(z0, z1, z2) -> c55 A__U82(tt, z0, z1) -> c56(A__U83(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U82(z0, z1, z2) -> c57 A__U83(tt, z0, z1) -> c58(A__U84(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U83(z0, z1, z2) -> c59 A__U84(tt, z0, z1) -> c60(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U84(tt, z0, z1) -> c61(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U84(z0, z1, z2) -> c62 A__U91(tt, z0) -> c63(A__U92(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U91(z0, z1) -> c64 A__U92(tt) -> c65 A__U92(z0) -> c66 A__ISNAT(0) -> c67 A__ISNAT(plus(z0, z1)) -> c68(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c69(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(x(z0, z1)) -> c70(A__U31(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(z0) -> c71 A__ISNATKIND(0) -> c72 A__ISNATKIND(plus(z0, z1)) -> c73(A__U41(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c74(A__U51(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(x(z0, z1)) -> c75(A__U61(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c76 A__PLUS(z0, 0) -> c77(A__U71(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c78(A__U81(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c79 A__X(z0, 0) -> c80(A__U91(a__isNat(z0), z0), A__ISNAT(z0)) A__X(z0, s(z1)) -> c81(A__U101(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c82 MARK(U101(z0, z1, z2)) -> c83(A__U101(mark(z0), z1, z2), MARK(z0)) MARK(U102(z0, z1, z2)) -> c84(A__U102(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c85(A__ISNATKIND(z0)) MARK(U103(z0, z1, z2)) -> c86(A__U103(mark(z0), z1, z2), MARK(z0)) MARK(isNat(z0)) -> c87(A__ISNAT(z0)) MARK(U104(z0, z1, z2)) -> c88(A__U104(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c89(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c90(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(x(z0, z1)) -> c91(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c92(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(U11(z0, z1, z2)) -> c93(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c94(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(U13(z0, z1, z2)) -> c95(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c96(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c97(A__U15(mark(z0), z1), MARK(z0)) MARK(U16(z0)) -> c98(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c99(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c100(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c101(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1, z2)) -> c102(A__U31(mark(z0), z1, z2), MARK(z0)) MARK(U32(z0, z1, z2)) -> c103(A__U32(mark(z0), z1, z2), MARK(z0)) MARK(U33(z0, z1, z2)) -> c104(A__U33(mark(z0), z1, z2), MARK(z0)) MARK(U34(z0, z1, z2)) -> c105(A__U34(mark(z0), z1, z2), MARK(z0)) MARK(U35(z0, z1)) -> c106(A__U35(mark(z0), z1), MARK(z0)) MARK(U36(z0)) -> c107(A__U36(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c108(A__U41(mark(z0), z1), MARK(z0)) MARK(U42(z0)) -> c109(A__U42(mark(z0)), MARK(z0)) MARK(U51(z0)) -> c110(A__U51(mark(z0)), MARK(z0)) MARK(U61(z0, z1)) -> c111(A__U61(mark(z0), z1), MARK(z0)) MARK(U62(z0)) -> c112(A__U62(mark(z0)), MARK(z0)) MARK(U71(z0, z1)) -> c113(A__U71(mark(z0), z1), MARK(z0)) MARK(U72(z0, z1)) -> c114(A__U72(mark(z0), z1), MARK(z0)) MARK(U81(z0, z1, z2)) -> c115(A__U81(mark(z0), z1, z2), MARK(z0)) MARK(U82(z0, z1, z2)) -> c116(A__U82(mark(z0), z1, z2), MARK(z0)) MARK(U83(z0, z1, z2)) -> c117(A__U83(mark(z0), z1, z2), MARK(z0)) MARK(U84(z0, z1, z2)) -> c118(A__U84(mark(z0), z1, z2), MARK(z0)) MARK(U91(z0, z1)) -> c119(A__U91(mark(z0), z1), MARK(z0)) MARK(U92(z0)) -> c120(A__U92(mark(z0)), MARK(z0)) MARK(tt) -> c121 MARK(s(z0)) -> c122(MARK(z0)) MARK(0) -> c123 The (relative) TRS S consists of the following rules: a__U101(tt, z0, z1) -> a__U102(a__isNatKind(z0), z0, z1) a__U101(z0, z1, z2) -> U101(z0, z1, z2) a__U102(tt, z0, z1) -> a__U103(a__isNat(z1), z0, z1) a__U102(z0, z1, z2) -> U102(z0, z1, z2) a__U103(tt, z0, z1) -> a__U104(a__isNatKind(z1), z0, z1) a__U103(z0, z1, z2) -> U103(z0, z1, z2) a__U104(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U104(z0, z1, z2) -> U104(z0, z1, z2) a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0, z1) -> a__U32(a__isNatKind(z0), z0, z1) a__U31(z0, z1, z2) -> U31(z0, z1, z2) a__U32(tt, z0, z1) -> a__U33(a__isNatKind(z1), z0, z1) a__U32(z0, z1, z2) -> U32(z0, z1, z2) a__U33(tt, z0, z1) -> a__U34(a__isNatKind(z1), z0, z1) a__U33(z0, z1, z2) -> U33(z0, z1, z2) a__U34(tt, z0, z1) -> a__U35(a__isNat(z0), z1) a__U34(z0, z1, z2) -> U34(z0, z1, z2) a__U35(tt, z0) -> a__U36(a__isNat(z0)) a__U35(z0, z1) -> U35(z0, z1) a__U36(tt) -> tt a__U36(z0) -> U36(z0) a__U41(tt, z0) -> a__U42(a__isNatKind(z0)) a__U41(z0, z1) -> U41(z0, z1) a__U42(tt) -> tt a__U42(z0) -> U42(z0) a__U51(tt) -> tt a__U51(z0) -> U51(z0) a__U61(tt, z0) -> a__U62(a__isNatKind(z0)) a__U61(z0, z1) -> U61(z0, z1) a__U62(tt) -> tt a__U62(z0) -> U62(z0) a__U71(tt, z0) -> a__U72(a__isNatKind(z0), z0) a__U71(z0, z1) -> U71(z0, z1) a__U72(tt, z0) -> mark(z0) a__U72(z0, z1) -> U72(z0, z1) a__U81(tt, z0, z1) -> a__U82(a__isNatKind(z0), z0, z1) a__U81(z0, z1, z2) -> U81(z0, z1, z2) a__U82(tt, z0, z1) -> a__U83(a__isNat(z1), z0, z1) a__U82(z0, z1, z2) -> U82(z0, z1, z2) a__U83(tt, z0, z1) -> a__U84(a__isNatKind(z1), z0, z1) a__U83(z0, z1, z2) -> U83(z0, z1, z2) a__U84(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U84(z0, z1, z2) -> U84(z0, z1, z2) a__U91(tt, z0) -> a__U92(a__isNatKind(z0)) a__U91(z0, z1) -> U91(z0, z1) a__U92(tt) -> 0 a__U92(z0) -> U92(z0) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(x(z0, z1)) -> a__U31(a__isNatKind(z0), z0, z1) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U41(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U51(a__isNatKind(z0)) a__isNatKind(x(z0, z1)) -> a__U61(a__isNatKind(z0), z1) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U71(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U81(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U91(a__isNat(z0), z0) a__x(z0, s(z1)) -> a__U101(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U101(z0, z1, z2)) -> a__U101(mark(z0), z1, z2) mark(U102(z0, z1, z2)) -> a__U102(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U103(z0, z1, z2)) -> a__U103(mark(z0), z1, z2) mark(isNat(z0)) -> a__isNat(z0) mark(U104(z0, z1, z2)) -> a__U104(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1, z2)) -> a__U31(mark(z0), z1, z2) mark(U32(z0, z1, z2)) -> a__U32(mark(z0), z1, z2) mark(U33(z0, z1, z2)) -> a__U33(mark(z0), z1, z2) mark(U34(z0, z1, z2)) -> a__U34(mark(z0), z1, z2) mark(U35(z0, z1)) -> a__U35(mark(z0), z1) mark(U36(z0)) -> a__U36(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U42(z0)) -> a__U42(mark(z0)) mark(U51(z0)) -> a__U51(mark(z0)) mark(U61(z0, z1)) -> a__U61(mark(z0), z1) mark(U62(z0)) -> a__U62(mark(z0)) mark(U71(z0, z1)) -> a__U71(mark(z0), z1) mark(U72(z0, z1)) -> a__U72(mark(z0), z1) mark(U81(z0, z1, z2)) -> a__U81(mark(z0), z1, z2) mark(U82(z0, z1, z2)) -> a__U82(mark(z0), z1, z2) mark(U83(z0, z1, z2)) -> a__U83(mark(z0), z1, z2) mark(U84(z0, z1, z2)) -> a__U84(mark(z0), z1, z2) mark(U91(z0, z1)) -> a__U91(mark(z0), z1) mark(U92(z0)) -> a__U92(mark(z0)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (8) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (9) BOUNDS(n^1, INF) ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__U101(tt, z0, z1) -> c(A__U102(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U101(z0, z1, z2) -> c1 A__U102(tt, z0, z1) -> c2(A__U103(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U102(z0, z1, z2) -> c3 A__U103(tt, z0, z1) -> c4(A__U104(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U103(z0, z1, z2) -> c5 A__U104(tt, z0, z1) -> c6(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U104(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U104(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U104(z0, z1, z2) -> c9 A__U11(tt, z0, z1) -> c10(A__U12(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U11(z0, z1, z2) -> c11 A__U12(tt, z0, z1) -> c12(A__U13(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U12(z0, z1, z2) -> c13 A__U13(tt, z0, z1) -> c14(A__U14(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U13(z0, z1, z2) -> c15 A__U14(tt, z0, z1) -> c16(A__U15(a__isNat(z0), z1), A__ISNAT(z0)) A__U14(z0, z1, z2) -> c17 A__U15(tt, z0) -> c18(A__U16(a__isNat(z0)), A__ISNAT(z0)) A__U15(z0, z1) -> c19 A__U16(tt) -> c20 A__U16(z0) -> c21 A__U21(tt, z0) -> c22(A__U22(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U21(z0, z1) -> c23 A__U22(tt, z0) -> c24(A__U23(a__isNat(z0)), A__ISNAT(z0)) A__U22(z0, z1) -> c25 A__U23(tt) -> c26 A__U23(z0) -> c27 A__U31(tt, z0, z1) -> c28(A__U32(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U31(z0, z1, z2) -> c29 A__U32(tt, z0, z1) -> c30(A__U33(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U32(z0, z1, z2) -> c31 A__U33(tt, z0, z1) -> c32(A__U34(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U33(z0, z1, z2) -> c33 A__U34(tt, z0, z1) -> c34(A__U35(a__isNat(z0), z1), A__ISNAT(z0)) A__U34(z0, z1, z2) -> c35 A__U35(tt, z0) -> c36(A__U36(a__isNat(z0)), A__ISNAT(z0)) A__U35(z0, z1) -> c37 A__U36(tt) -> c38 A__U36(z0) -> c39 A__U41(tt, z0) -> c40(A__U42(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U41(z0, z1) -> c41 A__U42(tt) -> c42 A__U42(z0) -> c43 A__U51(tt) -> c44 A__U51(z0) -> c45 A__U61(tt, z0) -> c46(A__U62(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U61(z0, z1) -> c47 A__U62(tt) -> c48 A__U62(z0) -> c49 A__U71(tt, z0) -> c50(A__U72(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__U71(z0, z1) -> c51 A__U72(tt, z0) -> c52(MARK(z0)) A__U72(z0, z1) -> c53 A__U81(tt, z0, z1) -> c54(A__U82(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__U81(z0, z1, z2) -> c55 A__U82(tt, z0, z1) -> c56(A__U83(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U82(z0, z1, z2) -> c57 A__U83(tt, z0, z1) -> c58(A__U84(a__isNatKind(z1), z0, z1), A__ISNATKIND(z1)) A__U83(z0, z1, z2) -> c59 A__U84(tt, z0, z1) -> c60(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U84(tt, z0, z1) -> c61(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U84(z0, z1, z2) -> c62 A__U91(tt, z0) -> c63(A__U92(a__isNatKind(z0)), A__ISNATKIND(z0)) A__U91(z0, z1) -> c64 A__U92(tt) -> c65 A__U92(z0) -> c66 A__ISNAT(0) -> c67 A__ISNAT(plus(z0, z1)) -> c68(A__U11(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c69(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(x(z0, z1)) -> c70(A__U31(a__isNatKind(z0), z0, z1), A__ISNATKIND(z0)) A__ISNAT(z0) -> c71 A__ISNATKIND(0) -> c72 A__ISNATKIND(plus(z0, z1)) -> c73(A__U41(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c74(A__U51(a__isNatKind(z0)), A__ISNATKIND(z0)) A__ISNATKIND(x(z0, z1)) -> c75(A__U61(a__isNatKind(z0), z1), A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c76 A__PLUS(z0, 0) -> c77(A__U71(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c78(A__U81(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c79 A__X(z0, 0) -> c80(A__U91(a__isNat(z0), z0), A__ISNAT(z0)) A__X(z0, s(z1)) -> c81(A__U101(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c82 MARK(U101(z0, z1, z2)) -> c83(A__U101(mark(z0), z1, z2), MARK(z0)) MARK(U102(z0, z1, z2)) -> c84(A__U102(mark(z0), z1, z2), MARK(z0)) MARK(isNatKind(z0)) -> c85(A__ISNATKIND(z0)) MARK(U103(z0, z1, z2)) -> c86(A__U103(mark(z0), z1, z2), MARK(z0)) MARK(isNat(z0)) -> c87(A__ISNAT(z0)) MARK(U104(z0, z1, z2)) -> c88(A__U104(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c89(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c90(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(x(z0, z1)) -> c91(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c92(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(U11(z0, z1, z2)) -> c93(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1, z2)) -> c94(A__U12(mark(z0), z1, z2), MARK(z0)) MARK(U13(z0, z1, z2)) -> c95(A__U13(mark(z0), z1, z2), MARK(z0)) MARK(U14(z0, z1, z2)) -> c96(A__U14(mark(z0), z1, z2), MARK(z0)) MARK(U15(z0, z1)) -> c97(A__U15(mark(z0), z1), MARK(z0)) MARK(U16(z0)) -> c98(A__U16(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c99(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0, z1)) -> c100(A__U22(mark(z0), z1), MARK(z0)) MARK(U23(z0)) -> c101(A__U23(mark(z0)), MARK(z0)) MARK(U31(z0, z1, z2)) -> c102(A__U31(mark(z0), z1, z2), MARK(z0)) MARK(U32(z0, z1, z2)) -> c103(A__U32(mark(z0), z1, z2), MARK(z0)) MARK(U33(z0, z1, z2)) -> c104(A__U33(mark(z0), z1, z2), MARK(z0)) MARK(U34(z0, z1, z2)) -> c105(A__U34(mark(z0), z1, z2), MARK(z0)) MARK(U35(z0, z1)) -> c106(A__U35(mark(z0), z1), MARK(z0)) MARK(U36(z0)) -> c107(A__U36(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c108(A__U41(mark(z0), z1), MARK(z0)) MARK(U42(z0)) -> c109(A__U42(mark(z0)), MARK(z0)) MARK(U51(z0)) -> c110(A__U51(mark(z0)), MARK(z0)) MARK(U61(z0, z1)) -> c111(A__U61(mark(z0), z1), MARK(z0)) MARK(U62(z0)) -> c112(A__U62(mark(z0)), MARK(z0)) MARK(U71(z0, z1)) -> c113(A__U71(mark(z0), z1), MARK(z0)) MARK(U72(z0, z1)) -> c114(A__U72(mark(z0), z1), MARK(z0)) MARK(U81(z0, z1, z2)) -> c115(A__U81(mark(z0), z1, z2), MARK(z0)) MARK(U82(z0, z1, z2)) -> c116(A__U82(mark(z0), z1, z2), MARK(z0)) MARK(U83(z0, z1, z2)) -> c117(A__U83(mark(z0), z1, z2), MARK(z0)) MARK(U84(z0, z1, z2)) -> c118(A__U84(mark(z0), z1, z2), MARK(z0)) MARK(U91(z0, z1)) -> c119(A__U91(mark(z0), z1), MARK(z0)) MARK(U92(z0)) -> c120(A__U92(mark(z0)), MARK(z0)) MARK(tt) -> c121 MARK(s(z0)) -> c122(MARK(z0)) MARK(0) -> c123 The (relative) TRS S consists of the following rules: a__U101(tt, z0, z1) -> a__U102(a__isNatKind(z0), z0, z1) a__U101(z0, z1, z2) -> U101(z0, z1, z2) a__U102(tt, z0, z1) -> a__U103(a__isNat(z1), z0, z1) a__U102(z0, z1, z2) -> U102(z0, z1, z2) a__U103(tt, z0, z1) -> a__U104(a__isNatKind(z1), z0, z1) a__U103(z0, z1, z2) -> U103(z0, z1, z2) a__U104(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U104(z0, z1, z2) -> U104(z0, z1, z2) a__U11(tt, z0, z1) -> a__U12(a__isNatKind(z0), z0, z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0, z1) -> a__U13(a__isNatKind(z1), z0, z1) a__U12(z0, z1, z2) -> U12(z0, z1, z2) a__U13(tt, z0, z1) -> a__U14(a__isNatKind(z1), z0, z1) a__U13(z0, z1, z2) -> U13(z0, z1, z2) a__U14(tt, z0, z1) -> a__U15(a__isNat(z0), z1) a__U14(z0, z1, z2) -> U14(z0, z1, z2) a__U15(tt, z0) -> a__U16(a__isNat(z0)) a__U15(z0, z1) -> U15(z0, z1) a__U16(tt) -> tt a__U16(z0) -> U16(z0) a__U21(tt, z0) -> a__U22(a__isNatKind(z0), z0) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt, z0) -> a__U23(a__isNat(z0)) a__U22(z0, z1) -> U22(z0, z1) a__U23(tt) -> tt a__U23(z0) -> U23(z0) a__U31(tt, z0, z1) -> a__U32(a__isNatKind(z0), z0, z1) a__U31(z0, z1, z2) -> U31(z0, z1, z2) a__U32(tt, z0, z1) -> a__U33(a__isNatKind(z1), z0, z1) a__U32(z0, z1, z2) -> U32(z0, z1, z2) a__U33(tt, z0, z1) -> a__U34(a__isNatKind(z1), z0, z1) a__U33(z0, z1, z2) -> U33(z0, z1, z2) a__U34(tt, z0, z1) -> a__U35(a__isNat(z0), z1) a__U34(z0, z1, z2) -> U34(z0, z1, z2) a__U35(tt, z0) -> a__U36(a__isNat(z0)) a__U35(z0, z1) -> U35(z0, z1) a__U36(tt) -> tt a__U36(z0) -> U36(z0) a__U41(tt, z0) -> a__U42(a__isNatKind(z0)) a__U41(z0, z1) -> U41(z0, z1) a__U42(tt) -> tt a__U42(z0) -> U42(z0) a__U51(tt) -> tt a__U51(z0) -> U51(z0) a__U61(tt, z0) -> a__U62(a__isNatKind(z0)) a__U61(z0, z1) -> U61(z0, z1) a__U62(tt) -> tt a__U62(z0) -> U62(z0) a__U71(tt, z0) -> a__U72(a__isNatKind(z0), z0) a__U71(z0, z1) -> U71(z0, z1) a__U72(tt, z0) -> mark(z0) a__U72(z0, z1) -> U72(z0, z1) a__U81(tt, z0, z1) -> a__U82(a__isNatKind(z0), z0, z1) a__U81(z0, z1, z2) -> U81(z0, z1, z2) a__U82(tt, z0, z1) -> a__U83(a__isNat(z1), z0, z1) a__U82(z0, z1, z2) -> U82(z0, z1, z2) a__U83(tt, z0, z1) -> a__U84(a__isNatKind(z1), z0, z1) a__U83(z0, z1, z2) -> U83(z0, z1, z2) a__U84(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U84(z0, z1, z2) -> U84(z0, z1, z2) a__U91(tt, z0) -> a__U92(a__isNatKind(z0)) a__U91(z0, z1) -> U91(z0, z1) a__U92(tt) -> 0 a__U92(z0) -> U92(z0) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNatKind(z0), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(x(z0, z1)) -> a__U31(a__isNatKind(z0), z0, z1) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__U41(a__isNatKind(z0), z1) a__isNatKind(s(z0)) -> a__U51(a__isNatKind(z0)) a__isNatKind(x(z0, z1)) -> a__U61(a__isNatKind(z0), z1) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U71(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U81(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U91(a__isNat(z0), z0) a__x(z0, s(z1)) -> a__U101(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U101(z0, z1, z2)) -> a__U101(mark(z0), z1, z2) mark(U102(z0, z1, z2)) -> a__U102(mark(z0), z1, z2) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(U103(z0, z1, z2)) -> a__U103(mark(z0), z1, z2) mark(isNat(z0)) -> a__isNat(z0) mark(U104(z0, z1, z2)) -> a__U104(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2) mark(U13(z0, z1, z2)) -> a__U13(mark(z0), z1, z2) mark(U14(z0, z1, z2)) -> a__U14(mark(z0), z1, z2) mark(U15(z0, z1)) -> a__U15(mark(z0), z1) mark(U16(z0)) -> a__U16(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0, z1)) -> a__U22(mark(z0), z1) mark(U23(z0)) -> a__U23(mark(z0)) mark(U31(z0, z1, z2)) -> a__U31(mark(z0), z1, z2) mark(U32(z0, z1, z2)) -> a__U32(mark(z0), z1, z2) mark(U33(z0, z1, z2)) -> a__U33(mark(z0), z1, z2) mark(U34(z0, z1, z2)) -> a__U34(mark(z0), z1, z2) mark(U35(z0, z1)) -> a__U35(mark(z0), z1) mark(U36(z0)) -> a__U36(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U42(z0)) -> a__U42(mark(z0)) mark(U51(z0)) -> a__U51(mark(z0)) mark(U61(z0, z1)) -> a__U61(mark(z0), z1) mark(U62(z0)) -> a__U62(mark(z0)) mark(U71(z0, z1)) -> a__U71(mark(z0), z1) mark(U72(z0, z1)) -> a__U72(mark(z0), z1) mark(U81(z0, z1, z2)) -> a__U81(mark(z0), z1, z2) mark(U82(z0, z1, z2)) -> a__U82(mark(z0), z1, z2) mark(U83(z0, z1, z2)) -> a__U83(mark(z0), z1, z2) mark(U84(z0, z1, z2)) -> a__U84(mark(z0), z1, z2) mark(U91(z0, z1)) -> a__U91(mark(z0), z1) mark(U92(z0)) -> a__U92(mark(z0)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST