/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 249 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: cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true p(0) -> 0 p(s(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(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond1(x_1, x_2, x_3, x_4)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond2(x_1, x_2, x_3, x_4)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond3(x_1, x_2, x_3, x_4)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3, x_4) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_cond2(x_1, x_2, x_3, x_4) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3, x_4) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(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: cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond1(x_1, x_2, x_3, x_4)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond2(x_1, x_2, x_3, x_4)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond3(x_1, x_2, x_3, x_4)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3, x_4) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_cond2(x_1, x_2, x_3, x_4) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3, x_4) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(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: cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond1(x_1, x_2, x_3, x_4)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond2(x_1, x_2, x_3, x_4)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond3(x_1, x_2, x_3, x_4)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3, x_4) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_cond2(x_1, x_2, x_3, x_4) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3, x_4) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(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: cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond1(x_1, x_2, x_3, x_4)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond2(x_1, x_2, x_3, x_4)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond3(x_1, x_2, x_3, x_4)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3, x_4) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_cond2(x_1, x_2, x_3, x_4) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3, x_4) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(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 gr(s(x), s(y)) ->^+ gr(x, y) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x), y / s(y)]. 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: cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond1(x_1, x_2, x_3, x_4)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond2(x_1, x_2, x_3, x_4)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond3(x_1, x_2, x_3, x_4)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3, x_4) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_cond2(x_1, x_2, x_3, x_4) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3, x_4) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(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: cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond1(x_1, x_2, x_3, x_4)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond2(x_1, x_2, x_3, x_4)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_cond3(x_1, x_2, x_3, x_4)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3, x_4) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_cond2(x_1, x_2, x_3, x_4) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3, x_4) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST