1131.61/291.56 WORST_CASE(Omega(n^1), ?) 1131.75/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1131.75/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1131.75/291.58 1131.75/291.58 1131.75/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.75/291.58 1131.75/291.58 (0) CpxTRS 1131.75/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1131.75/291.58 (2) TRS for Loop Detection 1131.75/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1131.75/291.58 (4) BEST 1131.75/291.58 (5) proven lower bound 1131.75/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1131.75/291.58 (7) BOUNDS(n^1, INF) 1131.75/291.58 (8) TRS for Loop Detection 1131.75/291.58 1131.75/291.58 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (0) 1131.75/291.58 Obligation: 1131.75/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.75/291.58 1131.75/291.58 1131.75/291.58 The TRS R consists of the following rules: 1131.75/291.58 1131.75/291.58 check(0) -> zero 1131.75/291.58 check(s(0)) -> odd 1131.75/291.58 check(s(s(0))) -> even 1131.75/291.58 check(s(s(s(x)))) -> check(s(x)) 1131.75/291.58 half(0) -> 0 1131.75/291.58 half(s(0)) -> 0 1131.75/291.58 half(s(s(x))) -> s(half(x)) 1131.75/291.58 plus(0, y) -> y 1131.75/291.58 plus(s(x), y) -> s(plus(x, y)) 1131.75/291.58 times(x, y) -> timesIter(x, y, 0) 1131.75/291.58 timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y)) 1131.75/291.58 p(s(x)) -> x 1131.75/291.58 p(0) -> 0 1131.75/291.58 if(zero, x, y, z, u) -> z 1131.75/291.58 if(odd, x, y, z, u) -> timesIter(p(x), y, u) 1131.75/291.58 if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z)))) 1131.75/291.58 1131.75/291.58 S is empty. 1131.75/291.58 Rewrite Strategy: INNERMOST 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1131.75/291.58 Transformed a relative TRS into a decreasing-loop problem. 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (2) 1131.75/291.58 Obligation: 1131.75/291.58 Analyzing the following TRS for decreasing loops: 1131.75/291.58 1131.75/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.75/291.58 1131.75/291.58 1131.75/291.58 The TRS R consists of the following rules: 1131.75/291.58 1131.75/291.58 check(0) -> zero 1131.75/291.58 check(s(0)) -> odd 1131.75/291.58 check(s(s(0))) -> even 1131.75/291.58 check(s(s(s(x)))) -> check(s(x)) 1131.75/291.58 half(0) -> 0 1131.75/291.58 half(s(0)) -> 0 1131.75/291.58 half(s(s(x))) -> s(half(x)) 1131.75/291.58 plus(0, y) -> y 1131.75/291.58 plus(s(x), y) -> s(plus(x, y)) 1131.75/291.58 times(x, y) -> timesIter(x, y, 0) 1131.75/291.58 timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y)) 1131.75/291.58 p(s(x)) -> x 1131.75/291.58 p(0) -> 0 1131.75/291.58 if(zero, x, y, z, u) -> z 1131.75/291.58 if(odd, x, y, z, u) -> timesIter(p(x), y, u) 1131.75/291.58 if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z)))) 1131.75/291.58 1131.75/291.58 S is empty. 1131.75/291.58 Rewrite Strategy: INNERMOST 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1131.75/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1131.75/291.58 1131.75/291.58 The rewrite sequence 1131.75/291.58 1131.75/291.58 check(s(s(s(x)))) ->^+ check(s(x)) 1131.75/291.58 1131.75/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1131.75/291.58 1131.75/291.58 The pumping substitution is [x / s(s(x))]. 1131.75/291.58 1131.75/291.58 The result substitution is [ ]. 1131.75/291.58 1131.75/291.58 1131.75/291.58 1131.75/291.58 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (4) 1131.75/291.58 Complex Obligation (BEST) 1131.75/291.58 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (5) 1131.75/291.58 Obligation: 1131.75/291.58 Proved the lower bound n^1 for the following obligation: 1131.75/291.58 1131.75/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.75/291.58 1131.75/291.58 1131.75/291.58 The TRS R consists of the following rules: 1131.75/291.58 1131.75/291.58 check(0) -> zero 1131.75/291.58 check(s(0)) -> odd 1131.75/291.58 check(s(s(0))) -> even 1131.75/291.58 check(s(s(s(x)))) -> check(s(x)) 1131.75/291.58 half(0) -> 0 1131.75/291.58 half(s(0)) -> 0 1131.75/291.58 half(s(s(x))) -> s(half(x)) 1131.75/291.58 plus(0, y) -> y 1131.75/291.58 plus(s(x), y) -> s(plus(x, y)) 1131.75/291.58 times(x, y) -> timesIter(x, y, 0) 1131.75/291.58 timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y)) 1131.75/291.58 p(s(x)) -> x 1131.75/291.58 p(0) -> 0 1131.75/291.58 if(zero, x, y, z, u) -> z 1131.75/291.58 if(odd, x, y, z, u) -> timesIter(p(x), y, u) 1131.75/291.58 if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z)))) 1131.75/291.58 1131.75/291.58 S is empty. 1131.75/291.58 Rewrite Strategy: INNERMOST 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (6) LowerBoundPropagationProof (FINISHED) 1131.75/291.58 Propagated lower bound. 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (7) 1131.75/291.58 BOUNDS(n^1, INF) 1131.75/291.58 1131.75/291.58 ---------------------------------------- 1131.75/291.58 1131.75/291.58 (8) 1131.75/291.58 Obligation: 1131.75/291.58 Analyzing the following TRS for decreasing loops: 1131.75/291.58 1131.75/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1131.75/291.58 1131.75/291.58 1131.75/291.58 The TRS R consists of the following rules: 1131.75/291.58 1131.75/291.58 check(0) -> zero 1131.75/291.58 check(s(0)) -> odd 1131.75/291.58 check(s(s(0))) -> even 1131.75/291.58 check(s(s(s(x)))) -> check(s(x)) 1131.75/291.58 half(0) -> 0 1131.75/291.58 half(s(0)) -> 0 1131.75/291.58 half(s(s(x))) -> s(half(x)) 1131.75/291.58 plus(0, y) -> y 1131.75/291.58 plus(s(x), y) -> s(plus(x, y)) 1131.75/291.58 times(x, y) -> timesIter(x, y, 0) 1131.75/291.58 timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y)) 1131.75/291.58 p(s(x)) -> x 1131.75/291.58 p(0) -> 0 1131.75/291.58 if(zero, x, y, z, u) -> z 1131.75/291.58 if(odd, x, y, z, u) -> timesIter(p(x), y, u) 1131.75/291.58 if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z)))) 1131.75/291.58 1131.75/291.58 S is empty. 1131.75/291.58 Rewrite Strategy: INNERMOST 1131.91/291.66 EOF