/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 444 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 0 -> n__0 s(X) -> n__s(X) diff(X1, X2) -> n__diff(X1, X2) p(X) -> n__p(X) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(X) -> X S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(true) -> true encArg(false) -> false encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__diff(x_1, x_2)) -> n__diff(encArg(x_1), encArg(x_2)) encArg(n__p(x_1)) -> n__p(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_activate(x_1) -> activate(encArg(x_1)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__diff(x_1, x_2) -> n__diff(encArg(x_1), encArg(x_2)) encode_n__p(x_1) -> n__p(encArg(x_1)) ---------------------------------------- (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: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 0 -> n__0 s(X) -> n__s(X) diff(X1, X2) -> n__diff(X1, X2) p(X) -> n__p(X) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__diff(x_1, x_2)) -> n__diff(encArg(x_1), encArg(x_2)) encArg(n__p(x_1)) -> n__p(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_activate(x_1) -> activate(encArg(x_1)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__diff(x_1, x_2) -> n__diff(encArg(x_1), encArg(x_2)) encode_n__p(x_1) -> n__p(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) 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: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 0 -> n__0 s(X) -> n__s(X) diff(X1, X2) -> n__diff(X1, X2) p(X) -> n__p(X) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__diff(x_1, x_2)) -> n__diff(encArg(x_1), encArg(x_2)) encArg(n__p(x_1)) -> n__p(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_activate(x_1) -> activate(encArg(x_1)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__diff(x_1, x_2) -> n__diff(encArg(x_1), encArg(x_2)) encode_n__p(x_1) -> n__p(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) 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: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 0 -> n__0 s(X) -> n__s(X) diff(X1, X2) -> n__diff(X1, X2) p(X) -> n__p(X) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__diff(x_1, x_2)) -> n__diff(encArg(x_1), encArg(x_2)) encArg(n__p(x_1)) -> n__p(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_activate(x_1) -> activate(encArg(x_1)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__diff(x_1, x_2) -> n__diff(encArg(x_1), encArg(x_2)) encode_n__p(x_1) -> n__p(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence activate(n__p(X)) ->^+ p(activate(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__p(X)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) 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: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 0 -> n__0 s(X) -> n__s(X) diff(X1, X2) -> n__diff(X1, X2) p(X) -> n__p(X) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__diff(x_1, x_2)) -> n__diff(encArg(x_1), encArg(x_2)) encArg(n__p(x_1)) -> n__p(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_activate(x_1) -> activate(encArg(x_1)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__diff(x_1, x_2) -> n__diff(encArg(x_1), encArg(x_2)) encode_n__p(x_1) -> n__p(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) 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: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) diff(X, Y) -> if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 0 -> n__0 s(X) -> n__s(X) diff(X1, X2) -> n__diff(X1, X2) p(X) -> n__p(X) activate(n__0) -> 0 activate(n__s(X)) -> s(activate(X)) activate(n__diff(X1, X2)) -> diff(activate(X1), activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__diff(x_1, x_2)) -> n__diff(encArg(x_1), encArg(x_2)) encArg(n__p(x_1)) -> n__p(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_activate(x_1) -> activate(encArg(x_1)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__diff(x_1, x_2) -> n__diff(encArg(x_1), encArg(x_2)) encode_n__p(x_1) -> n__p(encArg(x_1)) Rewrite Strategy: INNERMOST