/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 193 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (6) BEST (7) proven lower bound (8) LowerBoundPropagationProof [FINISHED, 0 ms] (9) BOUNDS(n^1, INF) (10) TRS for Loop Detection ---------------------------------------- (0) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (4) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) Rewrite Strategy: INNERMOST ---------------------------------------- (5) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence app(Cons(x, xs), ys) ->^+ Cons(x, app(xs, ys)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [xs / Cons(x, xs)]. The result substitution is [ ]. ---------------------------------------- (6) Complex Obligation (BEST) ---------------------------------------- (7) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) Rewrite Strategy: INNERMOST ---------------------------------------- (8) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (9) BOUNDS(n^1, INF) ---------------------------------------- (10) 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: quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) quicksort(Cons(x, Nil)) -> Cons(x, Nil) quicksort(Nil) -> Nil partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) partLt(x, Nil) -> Nil partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) partGt(x, Nil) -> Nil app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) app(Nil, ys) -> ys notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) goal(xs) -> quicksort(xs) The (relative) TRS S consists of the following rules: <(S(x), S(y)) -> <(x, y) <(0, S(y)) -> True <(x, 0) -> False >(S(x), S(y)) -> >(x, y) >(0, y) -> False >(S(x), 0) -> True partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) Rewrite Strategy: INNERMOST