WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 612 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(MARK(z0)) A__U11(z0, z1) -> c1 A__U21(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U21(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U21(z0, z1, z2) -> c4 A__U31(tt) -> c5 A__U31(z0) -> c6 A__U41(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U41(tt, z0, z1) -> c9(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U41(z0, z1, z2) -> c10 A__AND(tt, z0) -> c11(MARK(z0)) A__AND(z0, z1) -> c12 A__ISNAT(0) -> c13 A__ISNAT(plus(z0, z1)) -> c14(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c16(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(z0) -> c17 A__PLUS(z0, 0) -> c18(A__U11(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c19(A__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c20 A__X(z0, 0) -> c21(A__U31(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c22(A__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__X(z0, z1) -> c23 MARK(U11(z0, z1)) -> c24(A__U11(mark(z0), z1), MARK(z0)) MARK(U21(z0, z1, z2)) -> c25(A__U21(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c26(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U31(z0)) -> c28(A__U31(mark(z0)), MARK(z0)) MARK(U41(z0, z1, z2)) -> c29(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c30(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c31(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c32(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(tt) -> c34 MARK(s(z0)) -> c35(MARK(z0)) MARK(0) -> c36 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> mark(z0) a__U11(z0, z1) -> U11(z0, z1) a__U21(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U21(z0, z1, z2) -> U21(z0, z1, z2) a__U31(tt) -> 0 a__U31(z0) -> U31(z0) a__U41(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) 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__and(a__isNat(z0), isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0, z1)) -> a__and(a__isNat(z0), isNat(z1)) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U11(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U31(a__isNat(z0)) a__x(z0, s(z1)) -> a__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U21(z0, z1, z2)) -> a__U21(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(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) -> c(MARK(z0)) A__U11(z0, z1) -> c1 A__U21(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U21(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U21(z0, z1, z2) -> c4 A__U31(tt) -> c5 A__U31(z0) -> c6 A__U41(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U41(tt, z0, z1) -> c9(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U41(z0, z1, z2) -> c10 A__AND(tt, z0) -> c11(MARK(z0)) A__AND(z0, z1) -> c12 A__ISNAT(0) -> c13 A__ISNAT(plus(z0, z1)) -> c14(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c16(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(z0) -> c17 A__PLUS(z0, 0) -> c18(A__U11(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c19(A__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c20 A__X(z0, 0) -> c21(A__U31(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c22(A__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__X(z0, z1) -> c23 MARK(U11(z0, z1)) -> c24(A__U11(mark(z0), z1), MARK(z0)) MARK(U21(z0, z1, z2)) -> c25(A__U21(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c26(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U31(z0)) -> c28(A__U31(mark(z0)), MARK(z0)) MARK(U41(z0, z1, z2)) -> c29(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c30(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c31(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c32(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(tt) -> c34 MARK(s(z0)) -> c35(MARK(z0)) MARK(0) -> c36 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> mark(z0) a__U11(z0, z1) -> U11(z0, z1) a__U21(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U21(z0, z1, z2) -> U21(z0, z1, z2) a__U31(tt) -> 0 a__U31(z0) -> U31(z0) a__U41(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) 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__and(a__isNat(z0), isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0, z1)) -> a__and(a__isNat(z0), isNat(z1)) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U11(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U31(a__isNat(z0)) a__x(z0, s(z1)) -> a__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U21(z0, z1, z2)) -> a__U21(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(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) -> c(MARK(z0)) A__U11(z0, z1) -> c1 A__U21(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U21(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U21(z0, z1, z2) -> c4 A__U31(tt) -> c5 A__U31(z0) -> c6 A__U41(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U41(tt, z0, z1) -> c9(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U41(z0, z1, z2) -> c10 A__AND(tt, z0) -> c11(MARK(z0)) A__AND(z0, z1) -> c12 A__ISNAT(0) -> c13 A__ISNAT(plus(z0, z1)) -> c14(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c16(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(z0) -> c17 A__PLUS(z0, 0) -> c18(A__U11(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c19(A__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c20 A__X(z0, 0) -> c21(A__U31(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c22(A__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__X(z0, z1) -> c23 MARK(U11(z0, z1)) -> c24(A__U11(mark(z0), z1), MARK(z0)) MARK(U21(z0, z1, z2)) -> c25(A__U21(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c26(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U31(z0)) -> c28(A__U31(mark(z0)), MARK(z0)) MARK(U41(z0, z1, z2)) -> c29(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c30(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c31(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c32(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(tt) -> c34 MARK(s(z0)) -> c35(MARK(z0)) MARK(0) -> c36 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> mark(z0) a__U11(z0, z1) -> U11(z0, z1) a__U21(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U21(z0, z1, z2) -> U21(z0, z1, z2) a__U31(tt) -> 0 a__U31(z0) -> U31(z0) a__U41(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) 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__and(a__isNat(z0), isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0, z1)) -> a__and(a__isNat(z0), isNat(z1)) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U11(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U31(a__isNat(z0)) a__x(z0, s(z1)) -> a__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U21(z0, z1, z2)) -> a__U21(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(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(U41(z0, z1, z2)) ->^+ c29(A__U41(mark(z0), z1, z2), MARK(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / U41(z0, z1, z2)]. 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(MARK(z0)) A__U11(z0, z1) -> c1 A__U21(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U21(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U21(z0, z1, z2) -> c4 A__U31(tt) -> c5 A__U31(z0) -> c6 A__U41(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U41(tt, z0, z1) -> c9(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U41(z0, z1, z2) -> c10 A__AND(tt, z0) -> c11(MARK(z0)) A__AND(z0, z1) -> c12 A__ISNAT(0) -> c13 A__ISNAT(plus(z0, z1)) -> c14(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c16(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(z0) -> c17 A__PLUS(z0, 0) -> c18(A__U11(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c19(A__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c20 A__X(z0, 0) -> c21(A__U31(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c22(A__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__X(z0, z1) -> c23 MARK(U11(z0, z1)) -> c24(A__U11(mark(z0), z1), MARK(z0)) MARK(U21(z0, z1, z2)) -> c25(A__U21(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c26(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U31(z0)) -> c28(A__U31(mark(z0)), MARK(z0)) MARK(U41(z0, z1, z2)) -> c29(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c30(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c31(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c32(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(tt) -> c34 MARK(s(z0)) -> c35(MARK(z0)) MARK(0) -> c36 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> mark(z0) a__U11(z0, z1) -> U11(z0, z1) a__U21(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U21(z0, z1, z2) -> U21(z0, z1, z2) a__U31(tt) -> 0 a__U31(z0) -> U31(z0) a__U41(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) 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__and(a__isNat(z0), isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0, z1)) -> a__and(a__isNat(z0), isNat(z1)) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U11(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U31(a__isNat(z0)) a__x(z0, s(z1)) -> a__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U21(z0, z1, z2)) -> a__U21(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(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) -> c(MARK(z0)) A__U11(z0, z1) -> c1 A__U21(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1)) A__U21(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0)) A__U21(z0, z1, z2) -> c4 A__U31(tt) -> c5 A__U31(z0) -> c6 A__U41(tt, z0, z1) -> c7(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z1)) A__U41(tt, z0, z1) -> c8(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), A__X(mark(z1), mark(z0)), MARK(z0)) A__U41(tt, z0, z1) -> c9(A__PLUS(a__x(mark(z1), mark(z0)), mark(z1)), MARK(z1)) A__U41(z0, z1, z2) -> c10 A__AND(tt, z0) -> c11(MARK(z0)) A__AND(z0, z1) -> c12 A__ISNAT(0) -> c13 A__ISNAT(plus(z0, z1)) -> c14(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0, z1)) -> c16(A__AND(a__isNat(z0), isNat(z1)), A__ISNAT(z0)) A__ISNAT(z0) -> c17 A__PLUS(z0, 0) -> c18(A__U11(a__isNat(z0), z0), A__ISNAT(z0)) A__PLUS(z0, s(z1)) -> c19(A__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__PLUS(z0, z1) -> c20 A__X(z0, 0) -> c21(A__U31(a__isNat(z0)), A__ISNAT(z0)) A__X(z0, s(z1)) -> c22(A__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0), A__AND(a__isNat(z1), isNat(z0)), A__ISNAT(z1)) A__X(z0, z1) -> c23 MARK(U11(z0, z1)) -> c24(A__U11(mark(z0), z1), MARK(z0)) MARK(U21(z0, z1, z2)) -> c25(A__U21(mark(z0), z1, z2), MARK(z0)) MARK(plus(z0, z1)) -> c26(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c27(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(U31(z0)) -> c28(A__U31(mark(z0)), MARK(z0)) MARK(U41(z0, z1, z2)) -> c29(A__U41(mark(z0), z1, z2), MARK(z0)) MARK(x(z0, z1)) -> c30(A__X(mark(z0), mark(z1)), MARK(z0)) MARK(x(z0, z1)) -> c31(A__X(mark(z0), mark(z1)), MARK(z1)) MARK(and(z0, z1)) -> c32(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(tt) -> c34 MARK(s(z0)) -> c35(MARK(z0)) MARK(0) -> c36 The (relative) TRS S consists of the following rules: a__U11(tt, z0) -> mark(z0) a__U11(z0, z1) -> U11(z0, z1) a__U21(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0))) a__U21(z0, z1, z2) -> U21(z0, z1, z2) a__U31(tt) -> 0 a__U31(z0) -> U31(z0) a__U41(tt, z0, z1) -> a__plus(a__x(mark(z1), mark(z0)), mark(z1)) 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__and(a__isNat(z0), isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0, z1)) -> a__and(a__isNat(z0), isNat(z1)) a__isNat(z0) -> isNat(z0) a__plus(z0, 0) -> a__U11(a__isNat(z0), z0) a__plus(z0, s(z1)) -> a__U21(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__plus(z0, z1) -> plus(z0, z1) a__x(z0, 0) -> a__U31(a__isNat(z0)) a__x(z0, s(z1)) -> a__U41(a__and(a__isNat(z1), isNat(z0)), z1, z0) a__x(z0, z1) -> x(z0, z1) mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U21(z0, z1, z2)) -> a__U21(mark(z0), z1, z2) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2) mark(x(z0, z1)) -> a__x(mark(z0), mark(z1)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST