/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 118 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: minus(n__0, Y) -> 0 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) geq(X, n__0) -> true geq(n__0, n__s(Y)) -> false geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) div(0, n__s(Y)) -> 0 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) 0 -> n__0 s(X) -> n__s(X) div(X1, X2) -> n__div(X1, X2) minus(X1, X2) -> n__minus(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__div(X1, X2)) -> div(activate(X1), X2) activate(n__minus(X1, X2)) -> minus(X1, X2) 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: minus(n__0, Y) -> 0 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) geq(X, n__0) -> true geq(n__0, n__s(Y)) -> false geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) div(0, n__s(Y)) -> 0 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) 0 -> n__0 s(X) -> n__s(X) div(X1, X2) -> n__div(X1, X2) minus(X1, X2) -> n__minus(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__div(X1, X2)) -> div(activate(X1), X2) activate(n__minus(X1, X2)) -> minus(X1, X2) 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: minus(n__0, Y) -> 0 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) geq(X, n__0) -> true geq(n__0, n__s(Y)) -> false geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) div(0, n__s(Y)) -> 0 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) 0 -> n__0 s(X) -> n__s(X) div(X1, X2) -> n__div(X1, X2) minus(X1, X2) -> n__minus(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__div(X1, X2)) -> div(activate(X1), X2) activate(n__minus(X1, X2)) -> minus(X1, X2) 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: minus(n__0, Y) -> 0 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) geq(X, n__0) -> true geq(n__0, n__s(Y)) -> false geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) div(0, n__s(Y)) -> 0 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) 0 -> n__0 s(X) -> n__s(X) div(X1, X2) -> n__div(X1, X2) minus(X1, X2) -> n__minus(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__div(X1, X2)) -> div(activate(X1), X2) activate(n__minus(X1, X2)) -> minus(X1, X2) 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__div(n__s(X1_0), n__s(Y2_1))) ->^+ if(geq(activate(X1_0), activate(Y2_1)), n__s(n__div(n__minus(activate(X1_0), activate(Y2_1)), n__s(activate(Y2_1)))), n__0) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [X1_0 / n__div(n__s(X1_0), n__s(Y2_1))]. The result substitution is [ ]. The rewrite sequence activate(n__div(n__s(X1_0), n__s(Y2_1))) ->^+ if(geq(activate(X1_0), activate(Y2_1)), n__s(n__div(n__minus(activate(X1_0), activate(Y2_1)), n__s(activate(Y2_1)))), n__0) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0,0]. The pumping substitution is [X1_0 / n__div(n__s(X1_0), n__s(Y2_1))]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)