310.25/291.45 WORST_CASE(Omega(n^1), ?) 310.34/291.45 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 310.34/291.45 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 310.34/291.45 310.34/291.45 310.34/291.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.34/291.45 310.34/291.45 (0) CpxTRS 310.34/291.45 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 310.34/291.45 (2) TRS for Loop Detection 310.34/291.45 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 310.34/291.45 (4) BEST 310.34/291.45 (5) proven lower bound 310.34/291.45 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 310.34/291.45 (7) BOUNDS(n^1, INF) 310.34/291.45 (8) TRS for Loop Detection 310.34/291.45 310.34/291.45 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (0) 310.34/291.45 Obligation: 310.34/291.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.34/291.45 310.34/291.45 310.34/291.45 The TRS R consists of the following rules: 310.34/291.45 310.34/291.45 times(x, 0) -> 0 310.34/291.45 times(x, s(y)) -> plus(times(x, y), x) 310.34/291.45 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 310.34/291.45 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 310.34/291.45 plus(zero, y) -> y 310.34/291.45 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 310.34/291.45 id(x) -> x 310.34/291.45 if(true, x, y) -> x 310.34/291.45 if(false, x, y) -> y 310.34/291.45 not(x) -> if(x, false, true) 310.34/291.45 gt(s(x), zero) -> true 310.34/291.45 gt(zero, y) -> false 310.34/291.45 gt(s(x), s(y)) -> gt(x, y) 310.34/291.45 310.34/291.45 S is empty. 310.34/291.45 Rewrite Strategy: FULL 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 310.34/291.45 Transformed a relative TRS into a decreasing-loop problem. 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (2) 310.34/291.45 Obligation: 310.34/291.45 Analyzing the following TRS for decreasing loops: 310.34/291.45 310.34/291.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.34/291.45 310.34/291.45 310.34/291.45 The TRS R consists of the following rules: 310.34/291.45 310.34/291.45 times(x, 0) -> 0 310.34/291.45 times(x, s(y)) -> plus(times(x, y), x) 310.34/291.45 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 310.34/291.45 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 310.34/291.45 plus(zero, y) -> y 310.34/291.45 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 310.34/291.45 id(x) -> x 310.34/291.45 if(true, x, y) -> x 310.34/291.45 if(false, x, y) -> y 310.34/291.45 not(x) -> if(x, false, true) 310.34/291.45 gt(s(x), zero) -> true 310.34/291.45 gt(zero, y) -> false 310.34/291.45 gt(s(x), s(y)) -> gt(x, y) 310.34/291.45 310.34/291.45 S is empty. 310.34/291.45 Rewrite Strategy: FULL 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (3) DecreasingLoopProof (LOWER BOUND(ID)) 310.34/291.45 The following loop(s) give(s) rise to the lower bound Omega(n^1): 310.34/291.45 310.34/291.45 The rewrite sequence 310.34/291.45 310.34/291.45 gt(s(x), s(y)) ->^+ gt(x, y) 310.34/291.45 310.34/291.45 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 310.34/291.45 310.34/291.45 The pumping substitution is [x / s(x), y / s(y)]. 310.34/291.45 310.34/291.45 The result substitution is [ ]. 310.34/291.45 310.34/291.45 310.34/291.45 310.34/291.45 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (4) 310.34/291.45 Complex Obligation (BEST) 310.34/291.45 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (5) 310.34/291.45 Obligation: 310.34/291.45 Proved the lower bound n^1 for the following obligation: 310.34/291.45 310.34/291.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.34/291.45 310.34/291.45 310.34/291.45 The TRS R consists of the following rules: 310.34/291.45 310.34/291.45 times(x, 0) -> 0 310.34/291.45 times(x, s(y)) -> plus(times(x, y), x) 310.34/291.45 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 310.34/291.45 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 310.34/291.45 plus(zero, y) -> y 310.34/291.45 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 310.34/291.45 id(x) -> x 310.34/291.45 if(true, x, y) -> x 310.34/291.45 if(false, x, y) -> y 310.34/291.45 not(x) -> if(x, false, true) 310.34/291.45 gt(s(x), zero) -> true 310.34/291.45 gt(zero, y) -> false 310.34/291.45 gt(s(x), s(y)) -> gt(x, y) 310.34/291.45 310.34/291.45 S is empty. 310.34/291.45 Rewrite Strategy: FULL 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (6) LowerBoundPropagationProof (FINISHED) 310.34/291.45 Propagated lower bound. 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (7) 310.34/291.45 BOUNDS(n^1, INF) 310.34/291.45 310.34/291.45 ---------------------------------------- 310.34/291.45 310.34/291.45 (8) 310.34/291.45 Obligation: 310.34/291.45 Analyzing the following TRS for decreasing loops: 310.34/291.45 310.34/291.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.34/291.45 310.34/291.45 310.34/291.45 The TRS R consists of the following rules: 310.34/291.45 310.34/291.45 times(x, 0) -> 0 310.34/291.45 times(x, s(y)) -> plus(times(x, y), x) 310.34/291.45 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 310.34/291.45 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 310.34/291.45 plus(zero, y) -> y 310.34/291.45 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 310.34/291.45 id(x) -> x 310.34/291.45 if(true, x, y) -> x 310.34/291.45 if(false, x, y) -> y 310.34/291.45 not(x) -> if(x, false, true) 310.34/291.45 gt(s(x), zero) -> true 310.34/291.45 gt(zero, y) -> false 310.34/291.45 gt(s(x), s(y)) -> gt(x, y) 310.34/291.45 310.34/291.45 S is empty. 310.34/291.45 Rewrite Strategy: FULL 310.37/291.48 EOF