/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 104 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: a__minus(0, Y) -> 0 a__minus(s(X), s(Y)) -> a__minus(X, Y) a__geq(X, 0) -> true a__geq(0, s(Y)) -> false a__geq(s(X), s(Y)) -> a__geq(X, Y) a__div(0, s(Y)) -> 0 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) mark(minus(X1, X2)) -> a__minus(X1, X2) mark(geq(X1, X2)) -> a__geq(X1, X2) mark(div(X1, X2)) -> a__div(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(true) -> true mark(false) -> false a__minus(X1, X2) -> minus(X1, X2) a__geq(X1, X2) -> geq(X1, X2) a__div(X1, X2) -> div(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) 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: a__minus(0, Y) -> 0 a__minus(s(X), s(Y)) -> a__minus(X, Y) a__geq(X, 0) -> true a__geq(0, s(Y)) -> false a__geq(s(X), s(Y)) -> a__geq(X, Y) a__div(0, s(Y)) -> 0 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) mark(minus(X1, X2)) -> a__minus(X1, X2) mark(geq(X1, X2)) -> a__geq(X1, X2) mark(div(X1, X2)) -> a__div(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(true) -> true mark(false) -> false a__minus(X1, X2) -> minus(X1, X2) a__geq(X1, X2) -> geq(X1, X2) a__div(X1, X2) -> div(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) 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 a__geq(s(X), s(Y)) ->^+ a__geq(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 [ ]. ---------------------------------------- (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: a__minus(0, Y) -> 0 a__minus(s(X), s(Y)) -> a__minus(X, Y) a__geq(X, 0) -> true a__geq(0, s(Y)) -> false a__geq(s(X), s(Y)) -> a__geq(X, Y) a__div(0, s(Y)) -> 0 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) mark(minus(X1, X2)) -> a__minus(X1, X2) mark(geq(X1, X2)) -> a__geq(X1, X2) mark(div(X1, X2)) -> a__div(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(true) -> true mark(false) -> false a__minus(X1, X2) -> minus(X1, X2) a__geq(X1, X2) -> geq(X1, X2) a__div(X1, X2) -> div(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) 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: a__minus(0, Y) -> 0 a__minus(s(X), s(Y)) -> a__minus(X, Y) a__geq(X, 0) -> true a__geq(0, s(Y)) -> false a__geq(s(X), s(Y)) -> a__geq(X, Y) a__div(0, s(Y)) -> 0 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) mark(minus(X1, X2)) -> a__minus(X1, X2) mark(geq(X1, X2)) -> a__geq(X1, X2) mark(div(X1, X2)) -> a__div(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(true) -> true mark(false) -> false a__minus(X1, X2) -> minus(X1, X2) a__geq(X1, X2) -> geq(X1, X2) a__div(X1, X2) -> div(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) 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))) ->^+ a__if(a__geq(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))) ->^+ a__if(a__geq(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)