/export/starexec/sandbox2/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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 ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(prod(X1, X2)) ->^+ a__prod(mark(X1), mark(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / prod(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) S is empty. Rewrite Strategy: INNERMOST