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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 783 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 2 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, z1) -> c(A__U12(a__isNat(z0), z1), A__ISNAT(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0) -> c2(A__U13(a__isNat(z0)), A__ISNAT(z0)) A__U12(z0, z1) -> c3 A__U13(tt) -> c4 A__U13(z0) -> c5 A__U21(tt, z0) -> c6(A__U22(a__isNat(z0)), A__ISNAT(z0)) A__U21(z0, z1) -> c7 A__U22(tt) -> c8 A__U22(z0) -> c9 A__U31(tt, z0) -> c10(MARK(z0)) A__U31(z0, z1) -> c11 A__U41(tt, z0, z1) -> c12(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c13(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U41(z0, z1, z2) -> c14 A__AND(tt, z0) -> c15(MARK(z0)) A__AND(z0, z1) -> c16 A__ISNAT(0) -> c17 A__ISNAT(plus(z0, z1)) -> c18(A__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1), A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c19(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c20 A__ISNATKIND(0) -> c21 A__ISNATKIND(plus(z0, z1)) -> c22(A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c23(A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c24 A__PLUS(z0, 0) -> c25(A__U31(a__and(a__isNat(z0), isNatKind(z0)), z0), A__AND(a__isNat(z0), isNatKind(z0)), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c26(A__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), A__AND(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), A__AND(a__isNat(z1), isNatKind(z1)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c27 MARK(U11(z0, z1, z2)) -> c28(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1)) -> c29(A__U12(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c30(A__ISNAT(z0)) MARK(U13(z0)) -> c31(A__U13(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c32(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0)) -> c33(A__U22(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c34(A__U31(mark(z0), z1), MARK(z0)) MARK(U41(z0, z1, z2)) -> c35(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c36(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c37(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c38(A__AND(mark(z0), z1), MARK(z0)) MARK(isNatKind(z0)) -> c39(A__ISNATKIND(z0)) MARK(tt) -> c40 MARK(s(z0)) -> c41(MARK(z0)) MARK(0) -> c42 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNat(z0), z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0) -> a__U13(a__isNat(z0)) a__U12(z0, z1) -> U12(z0, z1) a__U13(tt) -> tt a__U13(z0) -> U13(z0) a__U21(tt, z0) -> a__U22(a__isNat(z0)) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt) -> tt a__U22(z0) -> U22(z0) a__U31(tt, z0) -> mark(z0) a__U31(z0, z1) -> U31(z0, z1) a__U41(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U41(z0, z1, z2) -> U41(z0, z1, z2) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__and(a__isNatKind(z0), isNatKind(z1)) a__isNatKind(s(z0)) -> a__isNatKind(z0) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U31(a__and(a__isNat(z0), isNatKind(z0)), z0) a__plus(z0, s(z1)) -> a__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U13(z0)) -> a__U13(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0)) -> a__U22(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNatKind(z0)) -> a__isNatKind(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__U11(tt, z0, z1) -> c(A__U12(a__isNat(z0), z1), A__ISNAT(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0) -> c2(A__U13(a__isNat(z0)), A__ISNAT(z0)) A__U12(z0, z1) -> c3 A__U13(tt) -> c4 A__U13(z0) -> c5 A__U21(tt, z0) -> c6(A__U22(a__isNat(z0)), A__ISNAT(z0)) A__U21(z0, z1) -> c7 A__U22(tt) -> c8 A__U22(z0) -> c9 A__U31(tt, z0) -> c10(MARK(z0)) A__U31(z0, z1) -> c11 A__U41(tt, z0, z1) -> c12(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c13(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U41(z0, z1, z2) -> c14 A__AND(tt, z0) -> c15(MARK(z0)) A__AND(z0, z1) -> c16 A__ISNAT(0) -> c17 A__ISNAT(plus(z0, z1)) -> c18(A__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1), A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c19(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c20 A__ISNATKIND(0) -> c21 A__ISNATKIND(plus(z0, z1)) -> c22(A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c23(A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c24 A__PLUS(z0, 0) -> c25(A__U31(a__and(a__isNat(z0), isNatKind(z0)), z0), A__AND(a__isNat(z0), isNatKind(z0)), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c26(A__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), A__AND(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), A__AND(a__isNat(z1), isNatKind(z1)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c27 MARK(U11(z0, z1, z2)) -> c28(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1)) -> c29(A__U12(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c30(A__ISNAT(z0)) MARK(U13(z0)) -> c31(A__U13(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c32(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0)) -> c33(A__U22(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c34(A__U31(mark(z0), z1), MARK(z0)) MARK(U41(z0, z1, z2)) -> c35(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c36(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c37(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c38(A__AND(mark(z0), z1), MARK(z0)) MARK(isNatKind(z0)) -> c39(A__ISNATKIND(z0)) MARK(tt) -> c40 MARK(s(z0)) -> c41(MARK(z0)) MARK(0) -> c42 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNat(z0), z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0) -> a__U13(a__isNat(z0)) a__U12(z0, z1) -> U12(z0, z1) a__U13(tt) -> tt a__U13(z0) -> U13(z0) a__U21(tt, z0) -> a__U22(a__isNat(z0)) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt) -> tt a__U22(z0) -> U22(z0) a__U31(tt, z0) -> mark(z0) a__U31(z0, z1) -> U31(z0, z1) a__U41(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U41(z0, z1, z2) -> U41(z0, z1, z2) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__and(a__isNatKind(z0), isNatKind(z1)) a__isNatKind(s(z0)) -> a__isNatKind(z0) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U31(a__and(a__isNat(z0), isNatKind(z0)), z0) a__plus(z0, s(z1)) -> a__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U13(z0)) -> a__U13(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0)) -> a__U22(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNatKind(z0)) -> a__isNatKind(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__U11(tt, z0, z1) -> c(A__U12(a__isNat(z0), z1), A__ISNAT(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0) -> c2(A__U13(a__isNat(z0)), A__ISNAT(z0)) A__U12(z0, z1) -> c3 A__U13(tt) -> c4 A__U13(z0) -> c5 A__U21(tt, z0) -> c6(A__U22(a__isNat(z0)), A__ISNAT(z0)) A__U21(z0, z1) -> c7 A__U22(tt) -> c8 A__U22(z0) -> c9 A__U31(tt, z0) -> c10(MARK(z0)) A__U31(z0, z1) -> c11 A__U41(tt, z0, z1) -> c12(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c13(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U41(z0, z1, z2) -> c14 A__AND(tt, z0) -> c15(MARK(z0)) A__AND(z0, z1) -> c16 A__ISNAT(0) -> c17 A__ISNAT(plus(z0, z1)) -> c18(A__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1), A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c19(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c20 A__ISNATKIND(0) -> c21 A__ISNATKIND(plus(z0, z1)) -> c22(A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c23(A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c24 A__PLUS(z0, 0) -> c25(A__U31(a__and(a__isNat(z0), isNatKind(z0)), z0), A__AND(a__isNat(z0), isNatKind(z0)), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c26(A__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), A__AND(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), A__AND(a__isNat(z1), isNatKind(z1)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c27 MARK(U11(z0, z1, z2)) -> c28(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1)) -> c29(A__U12(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c30(A__ISNAT(z0)) MARK(U13(z0)) -> c31(A__U13(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c32(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0)) -> c33(A__U22(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c34(A__U31(mark(z0), z1), MARK(z0)) MARK(U41(z0, z1, z2)) -> c35(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c36(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c37(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c38(A__AND(mark(z0), z1), MARK(z0)) MARK(isNatKind(z0)) -> c39(A__ISNATKIND(z0)) MARK(tt) -> c40 MARK(s(z0)) -> c41(MARK(z0)) MARK(0) -> c42 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNat(z0), z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0) -> a__U13(a__isNat(z0)) a__U12(z0, z1) -> U12(z0, z1) a__U13(tt) -> tt a__U13(z0) -> U13(z0) a__U21(tt, z0) -> a__U22(a__isNat(z0)) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt) -> tt a__U22(z0) -> U22(z0) a__U31(tt, z0) -> mark(z0) a__U31(z0, z1) -> U31(z0, z1) a__U41(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U41(z0, z1, z2) -> U41(z0, z1, z2) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__and(a__isNatKind(z0), isNatKind(z1)) a__isNatKind(s(z0)) -> a__isNatKind(z0) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U31(a__and(a__isNat(z0), isNatKind(z0)), z0) a__plus(z0, s(z1)) -> a__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U13(z0)) -> a__U13(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0)) -> a__U22(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNatKind(z0)) -> a__isNatKind(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(plus(z0, z1)) ->^+ c37(A__PLUS(mark(z0), mark(z1)), MARK(z1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z1 / plus(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, z1) -> c(A__U12(a__isNat(z0), z1), A__ISNAT(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0) -> c2(A__U13(a__isNat(z0)), A__ISNAT(z0)) A__U12(z0, z1) -> c3 A__U13(tt) -> c4 A__U13(z0) -> c5 A__U21(tt, z0) -> c6(A__U22(a__isNat(z0)), A__ISNAT(z0)) A__U21(z0, z1) -> c7 A__U22(tt) -> c8 A__U22(z0) -> c9 A__U31(tt, z0) -> c10(MARK(z0)) A__U31(z0, z1) -> c11 A__U41(tt, z0, z1) -> c12(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c13(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U41(z0, z1, z2) -> c14 A__AND(tt, z0) -> c15(MARK(z0)) A__AND(z0, z1) -> c16 A__ISNAT(0) -> c17 A__ISNAT(plus(z0, z1)) -> c18(A__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1), A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c19(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c20 A__ISNATKIND(0) -> c21 A__ISNATKIND(plus(z0, z1)) -> c22(A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c23(A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c24 A__PLUS(z0, 0) -> c25(A__U31(a__and(a__isNat(z0), isNatKind(z0)), z0), A__AND(a__isNat(z0), isNatKind(z0)), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c26(A__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), A__AND(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), A__AND(a__isNat(z1), isNatKind(z1)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c27 MARK(U11(z0, z1, z2)) -> c28(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1)) -> c29(A__U12(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c30(A__ISNAT(z0)) MARK(U13(z0)) -> c31(A__U13(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c32(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0)) -> c33(A__U22(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c34(A__U31(mark(z0), z1), MARK(z0)) MARK(U41(z0, z1, z2)) -> c35(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c36(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c37(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c38(A__AND(mark(z0), z1), MARK(z0)) MARK(isNatKind(z0)) -> c39(A__ISNATKIND(z0)) MARK(tt) -> c40 MARK(s(z0)) -> c41(MARK(z0)) MARK(0) -> c42 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNat(z0), z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0) -> a__U13(a__isNat(z0)) a__U12(z0, z1) -> U12(z0, z1) a__U13(tt) -> tt a__U13(z0) -> U13(z0) a__U21(tt, z0) -> a__U22(a__isNat(z0)) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt) -> tt a__U22(z0) -> U22(z0) a__U31(tt, z0) -> mark(z0) a__U31(z0, z1) -> U31(z0, z1) a__U41(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U41(z0, z1, z2) -> U41(z0, z1, z2) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__and(a__isNatKind(z0), isNatKind(z1)) a__isNatKind(s(z0)) -> a__isNatKind(z0) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U31(a__and(a__isNat(z0), isNatKind(z0)), z0) a__plus(z0, s(z1)) -> a__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U13(z0)) -> a__U13(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0)) -> a__U22(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNatKind(z0)) -> a__isNatKind(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__U11(tt, z0, z1) -> c(A__U12(a__isNat(z0), z1), A__ISNAT(z0)) A__U11(z0, z1, z2) -> c1 A__U12(tt, z0) -> c2(A__U13(a__isNat(z0)), A__ISNAT(z0)) A__U12(z0, z1) -> c3 A__U13(tt) -> c4 A__U13(z0) -> c5 A__U21(tt, z0) -> c6(A__U22(a__isNat(z0)), A__ISNAT(z0)) A__U21(z0, z1) -> c7 A__U22(tt) -> c8 A__U22(z0) -> c9 A__U31(tt, z0) -> c10(MARK(z0)) A__U31(z0, z1) -> c11 A__U41(tt, z0, z1) -> c12(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c13(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U41(z0, z1, z2) -> c14 A__AND(tt, z0) -> c15(MARK(z0)) A__AND(z0, z1) -> c16 A__ISNAT(0) -> c17 A__ISNAT(plus(z0, z1)) -> c18(A__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1), A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNAT(s(z0)) -> c19(A__U21(a__isNatKind(z0), z0), A__ISNATKIND(z0)) A__ISNAT(z0) -> c20 A__ISNATKIND(0) -> c21 A__ISNATKIND(plus(z0, z1)) -> c22(A__AND(a__isNatKind(z0), isNatKind(z1)), A__ISNATKIND(z0)) A__ISNATKIND(s(z0)) -> c23(A__ISNATKIND(z0)) A__ISNATKIND(z0) -> c24 A__PLUS(z0, 0) -> c25(A__U31(a__and(a__isNat(z0), isNatKind(z0)), z0), A__AND(a__isNat(z0), isNatKind(z0)), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c26(A__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), A__AND(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), A__AND(a__isNat(z1), isNatKind(z1)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c27 MARK(U11(z0, z1, z2)) -> c28(A__U11(mark(z0), z1, z2), MARK(z0)) MARK(U12(z0, z1)) -> c29(A__U12(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c30(A__ISNAT(z0)) MARK(U13(z0)) -> c31(A__U13(mark(z0)), MARK(z0)) MARK(U21(z0, z1)) -> c32(A__U21(mark(z0), z1), MARK(z0)) MARK(U22(z0)) -> c33(A__U22(mark(z0)), MARK(z0)) MARK(U31(z0, z1)) -> c34(A__U31(mark(z0), z1), MARK(z0)) MARK(U41(z0, z1, z2)) -> c35(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c36(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c37(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c38(A__AND(mark(z0), z1), MARK(z0)) MARK(isNatKind(z0)) -> c39(A__ISNATKIND(z0)) MARK(tt) -> c40 MARK(s(z0)) -> c41(MARK(z0)) MARK(0) -> c42 The (relative) TRS S consists of the following rules: a__U11(tt, z0, z1) -> a__U12(a__isNat(z0), z1) a__U11(z0, z1, z2) -> U11(z0, z1, z2) a__U12(tt, z0) -> a__U13(a__isNat(z0)) a__U12(z0, z1) -> U12(z0, z1) a__U13(tt) -> tt a__U13(z0) -> U13(z0) a__U21(tt, z0) -> a__U22(a__isNat(z0)) a__U21(z0, z1) -> U21(z0, z1) a__U22(tt) -> tt a__U22(z0) -> U22(z0) a__U31(tt, z0) -> mark(z0) a__U31(z0, z1) -> U31(z0, z1) a__U41(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U41(z0, z1, z2) -> U41(z0, z1, z2) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(plus(z0, z1)) -> a__U11(a__and(a__isNatKind(z0), isNatKind(z1)), z0, z1) a__isNat(s(z0)) -> a__U21(a__isNatKind(z0), z0) a__isNat(z0) -> isNat(z0) a__isNatKind(0) -> tt a__isNatKind(plus(z0, z1)) -> a__and(a__isNatKind(z0), isNatKind(z1)) a__isNatKind(s(z0)) -> a__isNatKind(z0) a__isNatKind(z0) -> isNatKind(z0) a__plus(z0, 0) -> a__U31(a__and(a__isNat(z0), isNatKind(z0)), z0) a__plus(z0, s(z1)) -> a__U41(a__and(a__and(a__isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0) a__plus(z0, z1) -> plus(z0, z1) mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(U13(z0)) -> a__U13(mark(z0)) mark(U21(z0, z1)) -> a__U21(mark(z0), z1) mark(U22(z0)) -> a__U22(mark(z0)) mark(U31(z0, z1)) -> a__U31(mark(z0), z1) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNatKind(z0)) -> a__isNatKind(z0) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST