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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 5526 ms] (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__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0)) A__U11(z0, z1) -> c1 A__U12(tt) -> c2 A__U12(z0) -> c3 A__U21(tt) -> c4 A__U21(z0) -> c5 A__U31(tt, z0) -> c6(A__U32(a__isNat(z0)), A__ISNAT(z0)) A__U31(z0, z1) -> c7 A__U32(tt) -> c8 A__U32(z0) -> c9 A__U41(tt, z0) -> c10(MARK(z0)) A__U41(z0, z1) -> c11 A__U51(tt, z0, z1) -> c12(A__U52(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U51(z0, z1, z2) -> c13 A__U52(tt, z0, z1) -> c14(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U52(tt, z0, z1) -> c15(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U52(z0, z1, z2) -> c16 A__U61(tt) -> c17 A__U61(z0) -> c18 A__U71(tt, z0, z1) -> c19(A__U72(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U71(z0, z1, z2) -> c20 A__U72(tt, z0, z1) -> c21(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U72(tt, z0, z1) -> c22(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U72(tt, z0, z1) -> c23(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U72(z0, z1, z2) -> c24 A__ISNAT(0) -> c25 A__ISNAT(plus(z0, z1)) -> c26(A__U11(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c27(A__U21(a__isNat(z0)), A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c28(A__U31(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(z0) -> c29 A__PLUS(z0, 0) -> c30(A__U41(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c31(A__U51(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c32 A__X(z0, 0) -> c33(A__U61(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c34(A__U71(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c35 MARK(U11(z0, z1)) -> c36(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0)) -> c37(A__U12(mark(z0)), MARK(z0)) MARK(isNat(z0)) -> c38(A__ISNAT(z0)) MARK(U21(z0)) -> c39(A__U21(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c40(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c41(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c42(A__U41(mark(z0), z1), MARK(z0)) MARK(U51(z0, z1, z2)) -> c43(A__U51(mark(z0), z1, z2), MARK(z0)) MARK(U52(z0, z1, z2)) -> c44(A__U52(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c45(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c46(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U61(z0)) -> c47(A__U61(mark(z0)), MARK(z0)) MARK(U71(z0, z1, z2)) -> c48(A__U71(mark(z0), z1, z2), MARK(z0)) MARK(U72(z0, z1, z2)) -> c49(A__U72(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c50(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c51(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c52 MARK(s(z0)) -> c53(MARK(z0)) MARK(0) -> c54 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> a__U12(a__isNat(z0)) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt) -> tt a__U12(z0) -> U12(z0) a__U21(tt) -> tt a__U21(z0) -> U21(z0) a__U31(tt, z0) -> a__U32(a__isNat(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt, z0) -> mark(z0) a__U41(z0, z1) -> U41(z0, z1) a__U51(tt, z0, z1) -> a__U52(a__isNat(z1), z0, z1) a__U51(z0, z1, z2) -> U51(z0, z1, z2) a__U52(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U52(z0, z1, z2) -> U52(z0, z1, z2) a__U61(tt) -> 0 a__U61(z0) -> U61(z0) a__U71(tt, z0, z1) -> a__U72(a__isNat(z1), z0, z1) a__U71(z0, z1, z2) -> U71(z0, z1, z2) a__U72(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U72(z0, z1, z2) -> U72(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__isNat(x(z0, z1)) -> a__U31(a__isNat(z0), z1) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U41(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U51(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U61(a__isNat(z0)) a__x(z0, s(z1)) -> a__U71(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(isNat(z0)) -> a__isNat(z0) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U51(z0, z1, z2)) -> a__U51(mark(z0), z1, z2) mark(U52(z0, z1, z2)) -> a__U52(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U61(z0)) -> a__U61(mark(z0)) mark(U71(z0, z1, z2)) -> a__U71(mark(z0), z1, z2) mark(U72(z0, z1, z2)) -> a__U72(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) 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__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0)) A__U11(z0, z1) -> c1 A__U12(tt) -> c2 A__U12(z0) -> c3 A__U21(tt) -> c4 A__U21(z0) -> c5 A__U31(tt, z0) -> c6(A__U32(a__isNat(z0)), A__ISNAT(z0)) A__U31(z0, z1) -> c7 A__U32(tt) -> c8 A__U32(z0) -> c9 A__U41(tt, z0) -> c10(MARK(z0)) A__U41(z0, z1) -> c11 A__U51(tt, z0, z1) -> c12(A__U52(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U51(z0, z1, z2) -> c13 A__U52(tt, z0, z1) -> c14(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U52(tt, z0, z1) -> c15(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U52(z0, z1, z2) -> c16 A__U61(tt) -> c17 A__U61(z0) -> c18 A__U71(tt, z0, z1) -> c19(A__U72(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U71(z0, z1, z2) -> c20 A__U72(tt, z0, z1) -> c21(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U72(tt, z0, z1) -> c22(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U72(tt, z0, z1) -> c23(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U72(z0, z1, z2) -> c24 A__ISNAT(0) -> c25 A__ISNAT(plus(z0, z1)) -> c26(A__U11(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c27(A__U21(a__isNat(z0)), A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c28(A__U31(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(z0) -> c29 A__PLUS(z0, 0) -> c30(A__U41(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c31(A__U51(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c32 A__X(z0, 0) -> c33(A__U61(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c34(A__U71(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c35 MARK(U11(z0, z1)) -> c36(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0)) -> c37(A__U12(mark(z0)), MARK(z0)) MARK(isNat(z0)) -> c38(A__ISNAT(z0)) MARK(U21(z0)) -> c39(A__U21(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c40(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c41(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c42(A__U41(mark(z0), z1), MARK(z0)) MARK(U51(z0, z1, z2)) -> c43(A__U51(mark(z0), z1, z2), MARK(z0)) MARK(U52(z0, z1, z2)) -> c44(A__U52(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c45(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c46(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U61(z0)) -> c47(A__U61(mark(z0)), MARK(z0)) MARK(U71(z0, z1, z2)) -> c48(A__U71(mark(z0), z1, z2), MARK(z0)) MARK(U72(z0, z1, z2)) -> c49(A__U72(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c50(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c51(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c52 MARK(s(z0)) -> c53(MARK(z0)) MARK(0) -> c54 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> a__U12(a__isNat(z0)) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt) -> tt a__U12(z0) -> U12(z0) a__U21(tt) -> tt a__U21(z0) -> U21(z0) a__U31(tt, z0) -> a__U32(a__isNat(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt, z0) -> mark(z0) a__U41(z0, z1) -> U41(z0, z1) a__U51(tt, z0, z1) -> a__U52(a__isNat(z1), z0, z1) a__U51(z0, z1, z2) -> U51(z0, z1, z2) a__U52(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U52(z0, z1, z2) -> U52(z0, z1, z2) a__U61(tt) -> 0 a__U61(z0) -> U61(z0) a__U71(tt, z0, z1) -> a__U72(a__isNat(z1), z0, z1) a__U71(z0, z1, z2) -> U71(z0, z1, z2) a__U72(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U72(z0, z1, z2) -> U72(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__isNat(x(z0, z1)) -> a__U31(a__isNat(z0), z1) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U41(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U51(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U61(a__isNat(z0)) a__x(z0, s(z1)) -> a__U71(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(isNat(z0)) -> a__isNat(z0) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U51(z0, z1, z2)) -> a__U51(mark(z0), z1, z2) mark(U52(z0, z1, z2)) -> a__U52(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U61(z0)) -> a__U61(mark(z0)) mark(U71(z0, z1, z2)) -> a__U71(mark(z0), z1, z2) mark(U72(z0, z1, z2)) -> a__U72(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) 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__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0)) A__U11(z0, z1) -> c1 A__U12(tt) -> c2 A__U12(z0) -> c3 A__U21(tt) -> c4 A__U21(z0) -> c5 A__U31(tt, z0) -> c6(A__U32(a__isNat(z0)), A__ISNAT(z0)) A__U31(z0, z1) -> c7 A__U32(tt) -> c8 A__U32(z0) -> c9 A__U41(tt, z0) -> c10(MARK(z0)) A__U41(z0, z1) -> c11 A__U51(tt, z0, z1) -> c12(A__U52(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U51(z0, z1, z2) -> c13 A__U52(tt, z0, z1) -> c14(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U52(tt, z0, z1) -> c15(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U52(z0, z1, z2) -> c16 A__U61(tt) -> c17 A__U61(z0) -> c18 A__U71(tt, z0, z1) -> c19(A__U72(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U71(z0, z1, z2) -> c20 A__U72(tt, z0, z1) -> c21(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U72(tt, z0, z1) -> c22(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U72(tt, z0, z1) -> c23(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U72(z0, z1, z2) -> c24 A__ISNAT(0) -> c25 A__ISNAT(plus(z0, z1)) -> c26(A__U11(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c27(A__U21(a__isNat(z0)), A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c28(A__U31(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(z0) -> c29 A__PLUS(z0, 0) -> c30(A__U41(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c31(A__U51(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c32 A__X(z0, 0) -> c33(A__U61(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c34(A__U71(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c35 MARK(U11(z0, z1)) -> c36(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0)) -> c37(A__U12(mark(z0)), MARK(z0)) MARK(isNat(z0)) -> c38(A__ISNAT(z0)) MARK(U21(z0)) -> c39(A__U21(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c40(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c41(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c42(A__U41(mark(z0), z1), MARK(z0)) MARK(U51(z0, z1, z2)) -> c43(A__U51(mark(z0), z1, z2), MARK(z0)) MARK(U52(z0, z1, z2)) -> c44(A__U52(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c45(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c46(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U61(z0)) -> c47(A__U61(mark(z0)), MARK(z0)) MARK(U71(z0, z1, z2)) -> c48(A__U71(mark(z0), z1, z2), MARK(z0)) MARK(U72(z0, z1, z2)) -> c49(A__U72(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c50(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c51(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c52 MARK(s(z0)) -> c53(MARK(z0)) MARK(0) -> c54 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> a__U12(a__isNat(z0)) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt) -> tt a__U12(z0) -> U12(z0) a__U21(tt) -> tt a__U21(z0) -> U21(z0) a__U31(tt, z0) -> a__U32(a__isNat(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt, z0) -> mark(z0) a__U41(z0, z1) -> U41(z0, z1) a__U51(tt, z0, z1) -> a__U52(a__isNat(z1), z0, z1) a__U51(z0, z1, z2) -> U51(z0, z1, z2) a__U52(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U52(z0, z1, z2) -> U52(z0, z1, z2) a__U61(tt) -> 0 a__U61(z0) -> U61(z0) a__U71(tt, z0, z1) -> a__U72(a__isNat(z1), z0, z1) a__U71(z0, z1, z2) -> U71(z0, z1, z2) a__U72(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U72(z0, z1, z2) -> U72(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__isNat(x(z0, z1)) -> a__U31(a__isNat(z0), z1) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U41(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U51(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U61(a__isNat(z0)) a__x(z0, s(z1)) -> a__U71(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(isNat(z0)) -> a__isNat(z0) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U51(z0, z1, z2)) -> a__U51(mark(z0), z1, z2) mark(U52(z0, z1, z2)) -> a__U52(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U61(z0)) -> a__U61(mark(z0)) mark(U71(z0, z1, z2)) -> a__U71(mark(z0), z1, z2) mark(U72(z0, z1, z2)) -> a__U72(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) 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(U11(z0, z1)) ->^+ c36(A__U11(mark(z0), z1), MARK(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / U11(z0, z1)]. 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__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0)) A__U11(z0, z1) -> c1 A__U12(tt) -> c2 A__U12(z0) -> c3 A__U21(tt) -> c4 A__U21(z0) -> c5 A__U31(tt, z0) -> c6(A__U32(a__isNat(z0)), A__ISNAT(z0)) A__U31(z0, z1) -> c7 A__U32(tt) -> c8 A__U32(z0) -> c9 A__U41(tt, z0) -> c10(MARK(z0)) A__U41(z0, z1) -> c11 A__U51(tt, z0, z1) -> c12(A__U52(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U51(z0, z1, z2) -> c13 A__U52(tt, z0, z1) -> c14(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U52(tt, z0, z1) -> c15(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U52(z0, z1, z2) -> c16 A__U61(tt) -> c17 A__U61(z0) -> c18 A__U71(tt, z0, z1) -> c19(A__U72(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U71(z0, z1, z2) -> c20 A__U72(tt, z0, z1) -> c21(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U72(tt, z0, z1) -> c22(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U72(tt, z0, z1) -> c23(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U72(z0, z1, z2) -> c24 A__ISNAT(0) -> c25 A__ISNAT(plus(z0, z1)) -> c26(A__U11(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c27(A__U21(a__isNat(z0)), A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c28(A__U31(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(z0) -> c29 A__PLUS(z0, 0) -> c30(A__U41(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c31(A__U51(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c32 A__X(z0, 0) -> c33(A__U61(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c34(A__U71(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c35 MARK(U11(z0, z1)) -> c36(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0)) -> c37(A__U12(mark(z0)), MARK(z0)) MARK(isNat(z0)) -> c38(A__ISNAT(z0)) MARK(U21(z0)) -> c39(A__U21(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c40(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c41(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c42(A__U41(mark(z0), z1), MARK(z0)) MARK(U51(z0, z1, z2)) -> c43(A__U51(mark(z0), z1, z2), MARK(z0)) MARK(U52(z0, z1, z2)) -> c44(A__U52(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c45(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c46(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U61(z0)) -> c47(A__U61(mark(z0)), MARK(z0)) MARK(U71(z0, z1, z2)) -> c48(A__U71(mark(z0), z1, z2), MARK(z0)) MARK(U72(z0, z1, z2)) -> c49(A__U72(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c50(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c51(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c52 MARK(s(z0)) -> c53(MARK(z0)) MARK(0) -> c54 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> a__U12(a__isNat(z0)) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt) -> tt a__U12(z0) -> U12(z0) a__U21(tt) -> tt a__U21(z0) -> U21(z0) a__U31(tt, z0) -> a__U32(a__isNat(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt, z0) -> mark(z0) a__U41(z0, z1) -> U41(z0, z1) a__U51(tt, z0, z1) -> a__U52(a__isNat(z1), z0, z1) a__U51(z0, z1, z2) -> U51(z0, z1, z2) a__U52(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U52(z0, z1, z2) -> U52(z0, z1, z2) a__U61(tt) -> 0 a__U61(z0) -> U61(z0) a__U71(tt, z0, z1) -> a__U72(a__isNat(z1), z0, z1) a__U71(z0, z1, z2) -> U71(z0, z1, z2) a__U72(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U72(z0, z1, z2) -> U72(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__isNat(x(z0, z1)) -> a__U31(a__isNat(z0), z1) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U41(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U51(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U61(a__isNat(z0)) a__x(z0, s(z1)) -> a__U71(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(isNat(z0)) -> a__isNat(z0) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U51(z0, z1, z2)) -> a__U51(mark(z0), z1, z2) mark(U52(z0, z1, z2)) -> a__U52(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U61(z0)) -> a__U61(mark(z0)) mark(U71(z0, z1, z2)) -> a__U71(mark(z0), z1, z2) mark(U72(z0, z1, z2)) -> a__U72(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) 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__U11(tt, z0) -> c(A__U12(a__isNat(z0)), A__ISNAT(z0)) A__U11(z0, z1) -> c1 A__U12(tt) -> c2 A__U12(z0) -> c3 A__U21(tt) -> c4 A__U21(z0) -> c5 A__U31(tt, z0) -> c6(A__U32(a__isNat(z0)), A__ISNAT(z0)) A__U31(z0, z1) -> c7 A__U32(tt) -> c8 A__U32(z0) -> c9 A__U41(tt, z0) -> c10(MARK(z0)) A__U41(z0, z1) -> c11 A__U51(tt, z0, z1) -> c12(A__U52(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U51(z0, z1, z2) -> c13 A__U52(tt, z0, z1) -> c14(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U52(tt, z0, z1) -> c15(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U52(z0, z1, z2) -> c16 A__U61(tt) -> c17 A__U61(z0) -> c18 A__U71(tt, z0, z1) -> c19(A__U72(a__isNat(z1), z0, z1), A__ISNAT(z1)) A__U71(z0, z1, z2) -> c20 A__U72(tt, z0, z1) -> c21(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U72(tt, z0, z1) -> c22(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U72(tt, z0, z1) -> c23(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U72(z0, z1, z2) -> c24 A__ISNAT(0) -> c25 A__ISNAT(plus(z0, z1)) -> c26(A__U11(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c27(A__U21(a__isNat(z0)), A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c28(A__U31(a__isNat(z0), z1), A__ISNAT(z0)) A__ISNAT(z0) -> c29 A__PLUS(z0, 0) -> c30(A__U41(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c31(A__U51(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__PLUS(z0, z1) -> c32 A__X(z0, 0) -> c33(A__U61(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c34(A__U71(a__isNat(z1), z1, z0), A__ISNAT(z1)) A__X(z0, z1) -> c35 MARK(U11(z0, z1)) -> c36(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0)) -> c37(A__U12(mark(z0)), MARK(z0)) MARK(isNat(z0)) -> c38(A__ISNAT(z0)) MARK(U21(z0)) -> c39(A__U21(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c40(A__U31(mark(z0), z1), MARK(z0)) MARK(U32(z0)) -> c41(A__U32(mark(z0)), MARK(z0)) MARK(U41(z0, z1)) -> c42(A__U41(mark(z0), z1), MARK(z0)) MARK(U51(z0, z1, z2)) -> c43(A__U51(mark(z0), z1, z2), MARK(z0)) MARK(U52(z0, z1, z2)) -> c44(A__U52(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c45(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c46(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U61(z0)) -> c47(A__U61(mark(z0)), MARK(z0)) MARK(U71(z0, z1, z2)) -> c48(A__U71(mark(z0), z1, z2), MARK(z0)) MARK(U72(z0, z1, z2)) -> c49(A__U72(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c50(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c51(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(tt) -> c52 MARK(s(z0)) -> c53(MARK(z0)) MARK(0) -> c54 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> a__U12(a__isNat(z0)) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt) -> tt a__U12(z0) -> U12(z0) a__U21(tt) -> tt a__U21(z0) -> U21(z0) a__U31(tt, z0) -> a__U32(a__isNat(z0)) a__U31(z0, z1) -> U31(z0, z1) a__U32(tt) -> tt a__U32(z0) -> U32(z0) a__U41(tt, z0) -> mark(z0) a__U41(z0, z1) -> U41(z0, z1) a__U51(tt, z0, z1) -> a__U52(a__isNat(z1), z0, z1) a__U51(z0, z1, z2) -> U51(z0, z1, z2) a__U52(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U52(z0, z1, z2) -> U52(z0, z1, z2) a__U61(tt) -> 0 a__U61(z0) -> U61(z0) a__U71(tt, z0, z1) -> a__U72(a__isNat(z1), z0, z1) a__U71(z0, z1, z2) -> U71(z0, z1, z2) a__U72(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) a__U72(z0, z1, z2) -> U72(z0, z1, z2) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__isNat(z0), z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__isNat(x(z0, z1)) -> a__U31(a__isNat(z0), z1) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U41(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U51(a__isNat(z1), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U61(a__isNat(z0)) a__x(z0, s(z1)) -> a__U71(a__isNat(z1), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(isNat(z0)) -> a__isNat(z0) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U32(z0)) -> a__U32(mark(z0)) mark(U41(z0, z1)) -> a__U41(mark(z0), z1) mark(U51(z0, z1, z2)) -> a__U51(mark(z0), z1, z2) mark(U52(z0, z1, z2)) -> a__U52(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U61(z0)) -> a__U61(mark(z0)) mark(U71(z0, z1, z2)) -> a__U71(mark(z0), z1, z2) mark(U72(z0, z1, z2)) -> a__U72(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST