WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 211 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: subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) subsets(Nil) -> Cons(Nil, Nil) mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) mapconsapp(x, Nil, rest) -> rest notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> subsets(xs) The (relative) TRS S consists of the following rules: subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) 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: subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) subsets(Nil) -> Cons(Nil, Nil) mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) mapconsapp(x, Nil, rest) -> rest notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> subsets(xs) The (relative) TRS S consists of the following rules: subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) 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: subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) subsets(Nil) -> Cons(Nil, Nil) mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) mapconsapp(x, Nil, rest) -> rest notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> subsets(xs) The (relative) TRS S consists of the following rules: subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) 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 subsets(Cons(x, xs)) ->^+ subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) 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: subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) subsets(Nil) -> Cons(Nil, Nil) mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) mapconsapp(x, Nil, rest) -> rest notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> subsets(xs) The (relative) TRS S consists of the following rules: subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) 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: subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs)) subsets(Nil) -> Cons(Nil, Nil) mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest)) mapconsapp(x, Nil, rest) -> rest notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False goal(xs) -> subsets(xs) The (relative) TRS S consists of the following rules: subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs) Rewrite Strategy: INNERMOST