/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 357 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: minus_active(0, y) -> 0 mark(0) -> 0 minus_active(s(x), s(y)) -> minus_active(x, y) mark(s(x)) -> s(mark(x)) ge_active(x, 0) -> true mark(minus(x, y)) -> minus_active(x, y) ge_active(0, s(y)) -> false mark(ge(x, y)) -> ge_active(x, y) ge_active(s(x), s(y)) -> ge_active(x, y) mark(div(x, y)) -> div_active(mark(x), y) div_active(0, s(y)) -> 0 mark(if(x, y, z)) -> if_active(mark(x), y, z) div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0) if_active(true, x, y) -> mark(x) minus_active(x, y) -> minus(x, y) if_active(false, x, y) -> mark(y) ge_active(x, y) -> ge(x, y) if_active(x, y, z) -> if(x, y, z) div_active(x, y) -> div(x, y) 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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2)) encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2)) encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2)) encode_true -> true encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_false -> false encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (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: minus_active(0, y) -> 0 mark(0) -> 0 minus_active(s(x), s(y)) -> minus_active(x, y) mark(s(x)) -> s(mark(x)) ge_active(x, 0) -> true mark(minus(x, y)) -> minus_active(x, y) ge_active(0, s(y)) -> false mark(ge(x, y)) -> ge_active(x, y) ge_active(s(x), s(y)) -> ge_active(x, y) mark(div(x, y)) -> div_active(mark(x), y) div_active(0, s(y)) -> 0 mark(if(x, y, z)) -> if_active(mark(x), y, z) div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0) if_active(true, x, y) -> mark(x) minus_active(x, y) -> minus(x, y) if_active(false, x, y) -> mark(y) ge_active(x, y) -> ge(x, y) if_active(x, y, z) -> if(x, y, z) div_active(x, y) -> div(x, y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2)) encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2)) encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2)) encode_true -> true encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_false -> false encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) 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: minus_active(0, y) -> 0 mark(0) -> 0 minus_active(s(x), s(y)) -> minus_active(x, y) mark(s(x)) -> s(mark(x)) ge_active(x, 0) -> true mark(minus(x, y)) -> minus_active(x, y) ge_active(0, s(y)) -> false mark(ge(x, y)) -> ge_active(x, y) ge_active(s(x), s(y)) -> ge_active(x, y) mark(div(x, y)) -> div_active(mark(x), y) div_active(0, s(y)) -> 0 mark(if(x, y, z)) -> if_active(mark(x), y, z) div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0) if_active(true, x, y) -> mark(x) minus_active(x, y) -> minus(x, y) if_active(false, x, y) -> mark(y) ge_active(x, y) -> ge(x, y) if_active(x, y, z) -> if(x, y, z) div_active(x, y) -> div(x, y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2)) encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2)) encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2)) encode_true -> true encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_false -> false encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) 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: minus_active(0, y) -> 0 mark(0) -> 0 minus_active(s(x), s(y)) -> minus_active(x, y) mark(s(x)) -> s(mark(x)) ge_active(x, 0) -> true mark(minus(x, y)) -> minus_active(x, y) ge_active(0, s(y)) -> false mark(ge(x, y)) -> ge_active(x, y) ge_active(s(x), s(y)) -> ge_active(x, y) mark(div(x, y)) -> div_active(mark(x), y) div_active(0, s(y)) -> 0 mark(if(x, y, z)) -> if_active(mark(x), y, z) div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0) if_active(true, x, y) -> mark(x) minus_active(x, y) -> minus(x, y) if_active(false, x, y) -> mark(y) ge_active(x, y) -> ge(x, y) if_active(x, y, z) -> if(x, y, z) div_active(x, y) -> div(x, y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2)) encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2)) encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2)) encode_true -> true encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_false -> false encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) 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 mark(s(x)) ->^+ s(mark(x)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x / s(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: minus_active(0, y) -> 0 mark(0) -> 0 minus_active(s(x), s(y)) -> minus_active(x, y) mark(s(x)) -> s(mark(x)) ge_active(x, 0) -> true mark(minus(x, y)) -> minus_active(x, y) ge_active(0, s(y)) -> false mark(ge(x, y)) -> ge_active(x, y) ge_active(s(x), s(y)) -> ge_active(x, y) mark(div(x, y)) -> div_active(mark(x), y) div_active(0, s(y)) -> 0 mark(if(x, y, z)) -> if_active(mark(x), y, z) div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0) if_active(true, x, y) -> mark(x) minus_active(x, y) -> minus(x, y) if_active(false, x, y) -> mark(y) ge_active(x, y) -> ge(x, y) if_active(x, y, z) -> if(x, y, z) div_active(x, y) -> div(x, y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2)) encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2)) encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2)) encode_true -> true encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_false -> false encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) 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: minus_active(0, y) -> 0 mark(0) -> 0 minus_active(s(x), s(y)) -> minus_active(x, y) mark(s(x)) -> s(mark(x)) ge_active(x, 0) -> true mark(minus(x, y)) -> minus_active(x, y) ge_active(0, s(y)) -> false mark(ge(x, y)) -> ge_active(x, y) ge_active(s(x), s(y)) -> ge_active(x, y) mark(div(x, y)) -> div_active(mark(x), y) div_active(0, s(y)) -> 0 mark(if(x, y, z)) -> if_active(mark(x), y, z) div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0) if_active(true, x, y) -> mark(x) minus_active(x, y) -> minus(x, y) if_active(false, x, y) -> mark(y) ge_active(x, y) -> ge(x, y) if_active(x, y, z) -> if(x, y, z) div_active(x, y) -> div(x, y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_minus_active(x_1, x_2)) -> minus_active(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encArg(cons_ge_active(x_1, x_2)) -> ge_active(encArg(x_1), encArg(x_2)) encArg(cons_div_active(x_1, x_2)) -> div_active(encArg(x_1), encArg(x_2)) encArg(cons_if_active(x_1, x_2, x_3)) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) encode_minus_active(x_1, x_2) -> minus_active(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_ge_active(x_1, x_2) -> ge_active(encArg(x_1), encArg(x_2)) encode_true -> true encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_false -> false encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_div_active(x_1, x_2) -> div_active(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if_active(x_1, x_2, x_3) -> if_active(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST