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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) mark(f(X1, X2)) -> a__f(mark(X1), X2) mark(g(X)) -> g(mark(X)) a__f(X1, X2) -> f(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__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) mark(f(X1, X2)) -> a__f(mark(X1), X2) mark(g(X)) -> g(mark(X)) a__f(X1, X2) -> f(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(f(X1, X2)) ->^+ a__f(mark(X1), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / f(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__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) mark(f(X1, X2)) -> a__f(mark(X1), X2) mark(g(X)) -> g(mark(X)) a__f(X1, X2) -> f(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__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) mark(f(X1, X2)) -> a__f(mark(X1), X2) mark(g(X)) -> g(mark(X)) a__f(X1, X2) -> f(X1, X2) S is empty. Rewrite Strategy: INNERMOST