/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 302 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: zeros -> cons(0, n__zeros) U11(tt) -> tt U21(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil 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: zeros -> cons(0, n__zeros) U11(tt) -> tt U21(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil 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: zeros -> cons(0, n__zeros) U11(tt) -> tt U21(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil 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: zeros -> cons(0, n__zeros) U11(tt) -> tt U21(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNatIList(activate(V2))) U42(tt) -> tt U51(tt, V2) -> U52(isNatList(activate(V2))) U52(tt) -> tt U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) U62(tt, L) -> s(length(activate(L))) isNat(n__0) -> tt isNat(n__length(V1)) -> U11(isNatList(activate(V1))) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNatIList(V) -> U31(isNatList(activate(V))) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) length(nil) -> 0 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) zeros -> n__zeros 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil 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__length(n__cons(X11_0, X22_0))) ->^+ U61(isNatList(activate(X22_0)), activate(X22_0), activate(X11_0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. The result substitution is [ ]. The rewrite sequence activate(n__length(n__cons(X11_0, X22_0))) ->^+ U61(isNatList(activate(X22_0)), activate(X22_0), activate(X11_0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)