1099.73/291.54 WORST_CASE(Omega(n^1), ?) 1099.95/291.57 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1099.95/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1099.95/291.57 1099.95/291.57 1099.95/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1099.95/291.57 1099.95/291.57 (0) CpxTRS 1099.95/291.57 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1099.95/291.57 (2) TRS for Loop Detection 1099.95/291.57 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1099.95/291.57 (4) BEST 1099.95/291.57 (5) proven lower bound 1099.95/291.57 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1099.95/291.57 (7) BOUNDS(n^1, INF) 1099.95/291.57 (8) TRS for Loop Detection 1099.95/291.57 1099.95/291.57 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (0) 1099.95/291.57 Obligation: 1099.95/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1099.95/291.57 1099.95/291.57 1099.95/291.57 The TRS R consists of the following rules: 1099.95/291.57 1099.95/291.57 double(x) -> permute(x, x, a) 1099.95/291.57 permute(x, y, a) -> permute(isZero(x), x, b) 1099.95/291.57 permute(false, x, b) -> permute(ack(x, x), p(x), c) 1099.95/291.57 permute(true, x, b) -> 0 1099.95/291.57 permute(y, x, c) -> s(s(permute(x, y, a))) 1099.95/291.57 p(0) -> 0 1099.95/291.57 p(s(x)) -> x 1099.95/291.57 ack(0, x) -> plus(x, s(0)) 1099.95/291.57 ack(s(x), 0) -> ack(x, s(0)) 1099.95/291.57 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1099.95/291.57 plus(0, y) -> y 1099.95/291.57 plus(s(x), y) -> plus(x, s(y)) 1099.95/291.57 plus(x, s(s(y))) -> s(plus(s(x), y)) 1099.95/291.57 plus(x, s(0)) -> s(x) 1099.95/291.57 plus(x, 0) -> x 1099.95/291.57 isZero(0) -> true 1099.95/291.57 isZero(s(x)) -> false 1099.95/291.57 1099.95/291.57 S is empty. 1099.95/291.57 Rewrite Strategy: INNERMOST 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1099.95/291.57 Transformed a relative TRS into a decreasing-loop problem. 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (2) 1099.95/291.57 Obligation: 1099.95/291.57 Analyzing the following TRS for decreasing loops: 1099.95/291.57 1099.95/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1099.95/291.57 1099.95/291.57 1099.95/291.57 The TRS R consists of the following rules: 1099.95/291.57 1099.95/291.57 double(x) -> permute(x, x, a) 1099.95/291.57 permute(x, y, a) -> permute(isZero(x), x, b) 1099.95/291.57 permute(false, x, b) -> permute(ack(x, x), p(x), c) 1099.95/291.57 permute(true, x, b) -> 0 1099.95/291.57 permute(y, x, c) -> s(s(permute(x, y, a))) 1099.95/291.57 p(0) -> 0 1099.95/291.57 p(s(x)) -> x 1099.95/291.57 ack(0, x) -> plus(x, s(0)) 1099.95/291.57 ack(s(x), 0) -> ack(x, s(0)) 1099.95/291.57 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1099.95/291.57 plus(0, y) -> y 1099.95/291.57 plus(s(x), y) -> plus(x, s(y)) 1099.95/291.57 plus(x, s(s(y))) -> s(plus(s(x), y)) 1099.95/291.57 plus(x, s(0)) -> s(x) 1099.95/291.57 plus(x, 0) -> x 1099.95/291.57 isZero(0) -> true 1099.95/291.57 isZero(s(x)) -> false 1099.95/291.57 1099.95/291.57 S is empty. 1099.95/291.57 Rewrite Strategy: INNERMOST 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1099.95/291.57 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1099.95/291.57 1099.95/291.57 The rewrite sequence 1099.95/291.57 1099.95/291.57 plus(x, s(s(y))) ->^+ s(plus(s(x), y)) 1099.95/291.57 1099.95/291.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1099.95/291.57 1099.95/291.57 The pumping substitution is [y / s(s(y))]. 1099.95/291.57 1099.95/291.57 The result substitution is [x / s(x)]. 1099.95/291.57 1099.95/291.57 1099.95/291.57 1099.95/291.57 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (4) 1099.95/291.57 Complex Obligation (BEST) 1099.95/291.57 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (5) 1099.95/291.57 Obligation: 1099.95/291.57 Proved the lower bound n^1 for the following obligation: 1099.95/291.57 1099.95/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1099.95/291.57 1099.95/291.57 1099.95/291.57 The TRS R consists of the following rules: 1099.95/291.57 1099.95/291.57 double(x) -> permute(x, x, a) 1099.95/291.57 permute(x, y, a) -> permute(isZero(x), x, b) 1099.95/291.57 permute(false, x, b) -> permute(ack(x, x), p(x), c) 1099.95/291.57 permute(true, x, b) -> 0 1099.95/291.57 permute(y, x, c) -> s(s(permute(x, y, a))) 1099.95/291.57 p(0) -> 0 1099.95/291.57 p(s(x)) -> x 1099.95/291.57 ack(0, x) -> plus(x, s(0)) 1099.95/291.57 ack(s(x), 0) -> ack(x, s(0)) 1099.95/291.57 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1099.95/291.57 plus(0, y) -> y 1099.95/291.57 plus(s(x), y) -> plus(x, s(y)) 1099.95/291.57 plus(x, s(s(y))) -> s(plus(s(x), y)) 1099.95/291.57 plus(x, s(0)) -> s(x) 1099.95/291.57 plus(x, 0) -> x 1099.95/291.57 isZero(0) -> true 1099.95/291.57 isZero(s(x)) -> false 1099.95/291.57 1099.95/291.57 S is empty. 1099.95/291.57 Rewrite Strategy: INNERMOST 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (6) LowerBoundPropagationProof (FINISHED) 1099.95/291.57 Propagated lower bound. 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (7) 1099.95/291.57 BOUNDS(n^1, INF) 1099.95/291.57 1099.95/291.57 ---------------------------------------- 1099.95/291.57 1099.95/291.57 (8) 1099.95/291.57 Obligation: 1099.95/291.57 Analyzing the following TRS for decreasing loops: 1099.95/291.57 1099.95/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1099.95/291.57 1099.95/291.57 1099.95/291.57 The TRS R consists of the following rules: 1099.95/291.57 1099.95/291.57 double(x) -> permute(x, x, a) 1099.95/291.57 permute(x, y, a) -> permute(isZero(x), x, b) 1099.95/291.57 permute(false, x, b) -> permute(ack(x, x), p(x), c) 1099.95/291.57 permute(true, x, b) -> 0 1099.95/291.57 permute(y, x, c) -> s(s(permute(x, y, a))) 1099.95/291.57 p(0) -> 0 1099.95/291.57 p(s(x)) -> x 1099.95/291.57 ack(0, x) -> plus(x, s(0)) 1099.95/291.57 ack(s(x), 0) -> ack(x, s(0)) 1099.95/291.57 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1099.95/291.57 plus(0, y) -> y 1099.95/291.57 plus(s(x), y) -> plus(x, s(y)) 1099.95/291.57 plus(x, s(s(y))) -> s(plus(s(x), y)) 1099.95/291.57 plus(x, s(0)) -> s(x) 1099.95/291.57 plus(x, 0) -> x 1099.95/291.57 isZero(0) -> true 1099.95/291.57 isZero(s(x)) -> false 1099.95/291.57 1099.95/291.57 S is empty. 1099.95/291.57 Rewrite Strategy: INNERMOST 1099.95/291.64 EOF