309.41/291.48 WORST_CASE(Omega(n^1), ?) 309.41/291.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 309.41/291.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 309.41/291.49 309.41/291.49 309.41/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.41/291.49 309.41/291.49 (0) CpxTRS 309.41/291.49 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 309.41/291.49 (2) TRS for Loop Detection 309.41/291.49 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 309.41/291.49 (4) BEST 309.41/291.49 (5) proven lower bound 309.41/291.49 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 309.41/291.49 (7) BOUNDS(n^1, INF) 309.41/291.49 (8) TRS for Loop Detection 309.41/291.49 309.41/291.49 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (0) 309.41/291.49 Obligation: 309.41/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.41/291.49 309.41/291.49 309.41/291.49 The TRS R consists of the following rules: 309.41/291.49 309.41/291.49 min(0, y) -> 0 309.41/291.49 min(x, 0) -> 0 309.41/291.49 min(s(x), s(y)) -> s(min(x, y)) 309.41/291.49 max(0, y) -> y 309.41/291.49 max(x, 0) -> x 309.41/291.49 max(s(x), s(y)) -> s(max(x, y)) 309.41/291.49 twice(0) -> 0 309.41/291.49 twice(s(x)) -> s(s(twice(x))) 309.41/291.49 -(x, 0) -> x 309.41/291.49 -(s(x), s(y)) -> -(x, y) 309.41/291.49 p(s(x)) -> x 309.41/291.49 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 309.41/291.49 309.41/291.49 S is empty. 309.41/291.49 Rewrite Strategy: FULL 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 309.41/291.49 Transformed a relative TRS into a decreasing-loop problem. 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (2) 309.41/291.49 Obligation: 309.41/291.49 Analyzing the following TRS for decreasing loops: 309.41/291.49 309.41/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.41/291.49 309.41/291.49 309.41/291.49 The TRS R consists of the following rules: 309.41/291.49 309.41/291.49 min(0, y) -> 0 309.41/291.49 min(x, 0) -> 0 309.41/291.49 min(s(x), s(y)) -> s(min(x, y)) 309.41/291.49 max(0, y) -> y 309.41/291.49 max(x, 0) -> x 309.41/291.49 max(s(x), s(y)) -> s(max(x, y)) 309.41/291.49 twice(0) -> 0 309.41/291.49 twice(s(x)) -> s(s(twice(x))) 309.41/291.49 -(x, 0) -> x 309.41/291.49 -(s(x), s(y)) -> -(x, y) 309.41/291.49 p(s(x)) -> x 309.41/291.49 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 309.41/291.49 309.41/291.49 S is empty. 309.41/291.49 Rewrite Strategy: FULL 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (3) DecreasingLoopProof (LOWER BOUND(ID)) 309.41/291.49 The following loop(s) give(s) rise to the lower bound Omega(n^1): 309.41/291.49 309.41/291.49 The rewrite sequence 309.41/291.49 309.41/291.49 -(s(x), s(y)) ->^+ -(x, y) 309.41/291.49 309.41/291.49 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 309.41/291.49 309.41/291.49 The pumping substitution is [x / s(x), y / s(y)]. 309.41/291.49 309.41/291.49 The result substitution is [ ]. 309.41/291.49 309.41/291.49 309.41/291.49 309.41/291.49 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (4) 309.41/291.49 Complex Obligation (BEST) 309.41/291.49 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (5) 309.41/291.49 Obligation: 309.41/291.49 Proved the lower bound n^1 for the following obligation: 309.41/291.49 309.41/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.41/291.49 309.41/291.49 309.41/291.49 The TRS R consists of the following rules: 309.41/291.49 309.41/291.49 min(0, y) -> 0 309.41/291.49 min(x, 0) -> 0 309.41/291.49 min(s(x), s(y)) -> s(min(x, y)) 309.41/291.49 max(0, y) -> y 309.41/291.49 max(x, 0) -> x 309.41/291.49 max(s(x), s(y)) -> s(max(x, y)) 309.41/291.49 twice(0) -> 0 309.41/291.49 twice(s(x)) -> s(s(twice(x))) 309.41/291.49 -(x, 0) -> x 309.41/291.49 -(s(x), s(y)) -> -(x, y) 309.41/291.49 p(s(x)) -> x 309.41/291.49 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 309.41/291.49 309.41/291.49 S is empty. 309.41/291.49 Rewrite Strategy: FULL 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (6) LowerBoundPropagationProof (FINISHED) 309.41/291.49 Propagated lower bound. 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (7) 309.41/291.49 BOUNDS(n^1, INF) 309.41/291.49 309.41/291.49 ---------------------------------------- 309.41/291.49 309.41/291.49 (8) 309.41/291.49 Obligation: 309.41/291.49 Analyzing the following TRS for decreasing loops: 309.41/291.49 309.41/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.41/291.49 309.41/291.49 309.41/291.49 The TRS R consists of the following rules: 309.41/291.49 309.41/291.49 min(0, y) -> 0 309.41/291.49 min(x, 0) -> 0 309.41/291.49 min(s(x), s(y)) -> s(min(x, y)) 309.41/291.49 max(0, y) -> y 309.41/291.49 max(x, 0) -> x 309.41/291.49 max(s(x), s(y)) -> s(max(x, y)) 309.41/291.49 twice(0) -> 0 309.41/291.49 twice(s(x)) -> s(s(twice(x))) 309.41/291.49 -(x, 0) -> x 309.41/291.49 -(s(x), s(y)) -> -(x, y) 309.41/291.49 p(s(x)) -> x 309.41/291.49 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 309.41/291.49 309.41/291.49 S is empty. 309.41/291.49 Rewrite Strategy: FULL 309.41/291.53 EOF