/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 89 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_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: 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_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: FULL ---------------------------------------- (3) 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 [ ]. ---------------------------------------- (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_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: 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_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: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence mark(div(s(x1_0), s(y2_1))) ->^+ if_active(ge_active(mark(x1_0), y2_1), s(div(minus(mark(x1_0), y2_1), s(y2_1))), 0) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [x1_0 / div(s(x1_0), s(y2_1))]. The result substitution is [ ]. The rewrite sequence mark(div(s(x1_0), s(y2_1))) ->^+ if_active(ge_active(mark(x1_0), y2_1), s(div(minus(mark(x1_0), y2_1), s(y2_1))), 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 / div(s(x1_0), s(y2_1))]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)