305.58/291.54 WORST_CASE(Omega(n^1), ?) 305.58/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 305.58/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.58/291.58 305.58/291.58 305.58/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.58/291.58 305.58/291.58 (0) CpxTRS 305.58/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 305.58/291.58 (2) TRS for Loop Detection 305.58/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 305.58/291.58 (4) BEST 305.58/291.58 (5) proven lower bound 305.58/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 305.58/291.58 (7) BOUNDS(n^1, INF) 305.58/291.58 (8) TRS for Loop Detection 305.58/291.58 305.58/291.58 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (0) 305.58/291.58 Obligation: 305.58/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.58/291.58 305.58/291.58 305.58/291.58 The TRS R consists of the following rules: 305.58/291.58 305.58/291.58 lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) 305.58/291.58 lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) 305.58/291.58 if(true, x, y, z, u) -> z 305.58/291.58 if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) 305.58/291.58 if2(true, x, y, z, u) -> z 305.58/291.58 if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) 305.58/291.58 plus(0, y) -> y 305.58/291.58 plus(s(x), y) -> s(plus(x, y)) 305.58/291.58 times(x, y) -> ifTimes(ge(0, x), x, y) 305.58/291.58 ifTimes(true, x, y) -> 0 305.58/291.58 ifTimes(false, x, y) -> plus(y, times(y, p(x))) 305.58/291.58 p(s(x)) -> x 305.58/291.58 p(0) -> s(s(0)) 305.58/291.58 ge(x, 0) -> true 305.58/291.58 ge(0, s(y)) -> false 305.58/291.58 ge(s(x), s(y)) -> ge(x, y) 305.58/291.58 or(true, y) -> true 305.58/291.58 or(false, y) -> y 305.58/291.58 divisible(0, s(y)) -> true 305.58/291.58 divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) 305.58/291.58 div(x, y, 0) -> divisible(x, y) 305.58/291.58 div(0, y, s(z)) -> false 305.58/291.58 div(s(x), y, s(z)) -> div(x, y, z) 305.58/291.58 a -> b 305.58/291.58 a -> c 305.58/291.58 305.58/291.58 S is empty. 305.58/291.58 Rewrite Strategy: FULL 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 305.58/291.58 Transformed a relative TRS into a decreasing-loop problem. 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (2) 305.58/291.58 Obligation: 305.58/291.58 Analyzing the following TRS for decreasing loops: 305.58/291.58 305.58/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.58/291.58 305.58/291.58 305.58/291.58 The TRS R consists of the following rules: 305.58/291.58 305.58/291.58 lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) 305.58/291.58 lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) 305.58/291.58 if(true, x, y, z, u) -> z 305.58/291.58 if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) 305.58/291.58 if2(true, x, y, z, u) -> z 305.58/291.58 if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) 305.58/291.58 plus(0, y) -> y 305.58/291.58 plus(s(x), y) -> s(plus(x, y)) 305.58/291.58 times(x, y) -> ifTimes(ge(0, x), x, y) 305.58/291.58 ifTimes(true, x, y) -> 0 305.58/291.58 ifTimes(false, x, y) -> plus(y, times(y, p(x))) 305.58/291.58 p(s(x)) -> x 305.58/291.58 p(0) -> s(s(0)) 305.58/291.58 ge(x, 0) -> true 305.58/291.58 ge(0, s(y)) -> false 305.58/291.58 ge(s(x), s(y)) -> ge(x, y) 305.58/291.58 or(true, y) -> true 305.58/291.58 or(false, y) -> y 305.58/291.58 divisible(0, s(y)) -> true 305.58/291.58 divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) 305.58/291.58 div(x, y, 0) -> divisible(x, y) 305.58/291.58 div(0, y, s(z)) -> false 305.58/291.58 div(s(x), y, s(z)) -> div(x, y, z) 305.58/291.58 a -> b 305.58/291.58 a -> c 305.58/291.58 305.58/291.58 S is empty. 305.58/291.58 Rewrite Strategy: FULL 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 305.58/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 305.58/291.58 305.58/291.58 The rewrite sequence 305.58/291.58 305.58/291.58 div(s(x), y, s(z)) ->^+ div(x, y, z) 305.58/291.58 305.58/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 305.58/291.58 305.58/291.58 The pumping substitution is [x / s(x), z / s(z)]. 305.58/291.58 305.58/291.58 The result substitution is [ ]. 305.58/291.58 305.58/291.58 305.58/291.58 305.58/291.58 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (4) 305.58/291.58 Complex Obligation (BEST) 305.58/291.58 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (5) 305.58/291.58 Obligation: 305.58/291.58 Proved the lower bound n^1 for the following obligation: 305.58/291.58 305.58/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.58/291.58 305.58/291.58 305.58/291.58 The TRS R consists of the following rules: 305.58/291.58 305.58/291.58 lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) 305.58/291.58 lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) 305.58/291.58 if(true, x, y, z, u) -> z 305.58/291.58 if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) 305.58/291.58 if2(true, x, y, z, u) -> z 305.58/291.58 if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) 305.58/291.58 plus(0, y) -> y 305.58/291.58 plus(s(x), y) -> s(plus(x, y)) 305.58/291.58 times(x, y) -> ifTimes(ge(0, x), x, y) 305.58/291.58 ifTimes(true, x, y) -> 0 305.58/291.58 ifTimes(false, x, y) -> plus(y, times(y, p(x))) 305.58/291.58 p(s(x)) -> x 305.58/291.58 p(0) -> s(s(0)) 305.58/291.58 ge(x, 0) -> true 305.58/291.58 ge(0, s(y)) -> false 305.58/291.58 ge(s(x), s(y)) -> ge(x, y) 305.58/291.58 or(true, y) -> true 305.58/291.58 or(false, y) -> y 305.58/291.58 divisible(0, s(y)) -> true 305.58/291.58 divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) 305.58/291.58 div(x, y, 0) -> divisible(x, y) 305.58/291.58 div(0, y, s(z)) -> false 305.58/291.58 div(s(x), y, s(z)) -> div(x, y, z) 305.58/291.58 a -> b 305.58/291.58 a -> c 305.58/291.58 305.58/291.58 S is empty. 305.58/291.58 Rewrite Strategy: FULL 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (6) LowerBoundPropagationProof (FINISHED) 305.58/291.58 Propagated lower bound. 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (7) 305.58/291.58 BOUNDS(n^1, INF) 305.58/291.58 305.58/291.58 ---------------------------------------- 305.58/291.58 305.58/291.58 (8) 305.58/291.58 Obligation: 305.58/291.58 Analyzing the following TRS for decreasing loops: 305.58/291.58 305.58/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.58/291.58 305.58/291.58 305.58/291.58 The TRS R consists of the following rules: 305.58/291.58 305.58/291.58 lcm(x, y) -> lcmIter(x, y, 0, times(x, y)) 305.58/291.58 lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u) 305.58/291.58 if(true, x, y, z, u) -> z 305.58/291.58 if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u) 305.58/291.58 if2(true, x, y, z, u) -> z 305.58/291.58 if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u) 305.58/291.58 plus(0, y) -> y 305.58/291.58 plus(s(x), y) -> s(plus(x, y)) 305.58/291.58 times(x, y) -> ifTimes(ge(0, x), x, y) 305.58/291.58 ifTimes(true, x, y) -> 0 305.58/291.58 ifTimes(false, x, y) -> plus(y, times(y, p(x))) 305.58/291.58 p(s(x)) -> x 305.58/291.58 p(0) -> s(s(0)) 305.58/291.58 ge(x, 0) -> true 305.58/291.58 ge(0, s(y)) -> false 305.58/291.58 ge(s(x), s(y)) -> ge(x, y) 305.58/291.58 or(true, y) -> true 305.58/291.58 or(false, y) -> y 305.58/291.58 divisible(0, s(y)) -> true 305.58/291.58 divisible(s(x), s(y)) -> div(s(x), s(y), s(y)) 305.58/291.58 div(x, y, 0) -> divisible(x, y) 305.58/291.58 div(0, y, s(z)) -> false 305.58/291.58 div(s(x), y, s(z)) -> div(x, y, z) 305.58/291.58 a -> b 305.58/291.58 a -> c 305.58/291.58 305.58/291.58 S is empty. 305.58/291.58 Rewrite Strategy: FULL 305.70/291.61 EOF