/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 87 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U13(tt) -> tt U21(tt, V1) -> U22(isNat(activate(V1))) U22(tt) -> tt U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) U32(tt, V2) -> U33(isNat(activate(V2))) U33(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) and(X1, X2) -> n__and(X1, X2) isNat(X) -> n__isNat(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(n__and(X1, X2)) -> and(activate(X1), X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U13(tt) -> tt U21(tt, V1) -> U22(isNat(activate(V1))) U22(tt) -> tt U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) U32(tt, V2) -> U33(isNat(activate(V2))) U33(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) and(X1, X2) -> n__and(X1, X2) isNat(X) -> n__isNat(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(n__and(X1, X2)) -> and(activate(X1), X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence activate(n__s(X)) ->^+ s(activate(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__s(X)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U13(tt) -> tt U21(tt, V1) -> U22(isNat(activate(V1))) U22(tt) -> tt U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) U32(tt, V2) -> U33(isNat(activate(V2))) U33(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) and(X1, X2) -> n__and(X1, X2) isNat(X) -> n__isNat(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(n__and(X1, X2)) -> and(activate(X1), X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U13(tt) -> tt U21(tt, V1) -> U22(isNat(activate(V1))) U22(tt) -> tt U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) U32(tt, V2) -> U33(isNat(activate(V2))) U33(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0 U71(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) and(X1, X2) -> n__and(X1, X2) isNat(X) -> n__isNat(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(n__and(X1, X2)) -> and(activate(X1), X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence activate(n__isNat(n__x(V11_0, V22_0))) ->^+ U31(and(isNatKind(activate(V11_0)), n__isNatKind(activate(V22_0))), activate(V11_0), activate(V22_0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0]. The pumping substitution is [V11_0 / n__isNat(n__x(V11_0, V22_0))]. The result substitution is [ ]. The rewrite sequence activate(n__isNat(n__x(V11_0, V22_0))) ->^+ U31(and(isNatKind(activate(V11_0)), n__isNatKind(activate(V22_0))), activate(V11_0), activate(V22_0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [V11_0 / n__isNat(n__x(V11_0, V22_0))]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)