1114.55/291.57 WORST_CASE(Omega(n^1), ?) 1114.55/291.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1114.55/291.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1114.55/291.60 1114.55/291.60 1114.55/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.55/291.60 1114.55/291.60 (0) CpxTRS 1114.55/291.60 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1114.55/291.60 (2) TRS for Loop Detection 1114.55/291.60 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1114.55/291.60 (4) BEST 1114.55/291.60 (5) proven lower bound 1114.55/291.60 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1114.55/291.60 (7) BOUNDS(n^1, INF) 1114.55/291.60 (8) TRS for Loop Detection 1114.55/291.60 1114.55/291.60 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (0) 1114.55/291.60 Obligation: 1114.55/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.55/291.60 1114.55/291.60 1114.55/291.60 The TRS R consists of the following rules: 1114.55/291.60 1114.55/291.60 times(x, y) -> help(x, y, 0) 1114.55/291.60 help(x, y, c) -> if(lt(c, y), x, y, c) 1114.55/291.60 if(true, x, y, c) -> plus(x, help(x, y, s(c))) 1114.55/291.60 if(false, x, y, c) -> 0 1114.55/291.60 lt(0, s(x)) -> true 1114.55/291.60 lt(s(x), 0) -> false 1114.55/291.60 lt(s(x), s(y)) -> lt(x, y) 1114.55/291.60 plus(x, 0) -> x 1114.55/291.60 plus(0, x) -> x 1114.55/291.60 plus(x, s(y)) -> s(plus(x, y)) 1114.55/291.60 plus(s(x), y) -> s(plus(x, y)) 1114.55/291.60 1114.55/291.60 S is empty. 1114.55/291.60 Rewrite Strategy: FULL 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1114.55/291.60 Transformed a relative TRS into a decreasing-loop problem. 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (2) 1114.55/291.60 Obligation: 1114.55/291.60 Analyzing the following TRS for decreasing loops: 1114.55/291.60 1114.55/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.55/291.60 1114.55/291.60 1114.55/291.60 The TRS R consists of the following rules: 1114.55/291.60 1114.55/291.60 times(x, y) -> help(x, y, 0) 1114.55/291.60 help(x, y, c) -> if(lt(c, y), x, y, c) 1114.55/291.60 if(true, x, y, c) -> plus(x, help(x, y, s(c))) 1114.55/291.60 if(false, x, y, c) -> 0 1114.55/291.60 lt(0, s(x)) -> true 1114.55/291.60 lt(s(x), 0) -> false 1114.55/291.60 lt(s(x), s(y)) -> lt(x, y) 1114.55/291.60 plus(x, 0) -> x 1114.55/291.60 plus(0, x) -> x 1114.55/291.60 plus(x, s(y)) -> s(plus(x, y)) 1114.55/291.60 plus(s(x), y) -> s(plus(x, y)) 1114.55/291.60 1114.55/291.60 S is empty. 1114.55/291.60 Rewrite Strategy: FULL 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1114.55/291.60 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1114.55/291.60 1114.55/291.60 The rewrite sequence 1114.55/291.60 1114.55/291.60 lt(s(x), s(y)) ->^+ lt(x, y) 1114.55/291.60 1114.55/291.60 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1114.55/291.60 1114.55/291.60 The pumping substitution is [x / s(x), y / s(y)]. 1114.55/291.60 1114.55/291.60 The result substitution is [ ]. 1114.55/291.60 1114.55/291.60 1114.55/291.60 1114.55/291.60 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (4) 1114.55/291.60 Complex Obligation (BEST) 1114.55/291.60 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (5) 1114.55/291.60 Obligation: 1114.55/291.60 Proved the lower bound n^1 for the following obligation: 1114.55/291.60 1114.55/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.55/291.60 1114.55/291.60 1114.55/291.60 The TRS R consists of the following rules: 1114.55/291.60 1114.55/291.60 times(x, y) -> help(x, y, 0) 1114.55/291.60 help(x, y, c) -> if(lt(c, y), x, y, c) 1114.55/291.60 if(true, x, y, c) -> plus(x, help(x, y, s(c))) 1114.55/291.60 if(false, x, y, c) -> 0 1114.55/291.60 lt(0, s(x)) -> true 1114.55/291.60 lt(s(x), 0) -> false 1114.55/291.60 lt(s(x), s(y)) -> lt(x, y) 1114.55/291.60 plus(x, 0) -> x 1114.55/291.60 plus(0, x) -> x 1114.55/291.60 plus(x, s(y)) -> s(plus(x, y)) 1114.55/291.60 plus(s(x), y) -> s(plus(x, y)) 1114.55/291.60 1114.55/291.60 S is empty. 1114.55/291.60 Rewrite Strategy: FULL 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (6) LowerBoundPropagationProof (FINISHED) 1114.55/291.60 Propagated lower bound. 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (7) 1114.55/291.60 BOUNDS(n^1, INF) 1114.55/291.60 1114.55/291.60 ---------------------------------------- 1114.55/291.60 1114.55/291.60 (8) 1114.55/291.60 Obligation: 1114.55/291.60 Analyzing the following TRS for decreasing loops: 1114.55/291.60 1114.55/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1114.55/291.60 1114.55/291.60 1114.55/291.60 The TRS R consists of the following rules: 1114.55/291.60 1114.55/291.60 times(x, y) -> help(x, y, 0) 1114.55/291.60 help(x, y, c) -> if(lt(c, y), x, y, c) 1114.55/291.60 if(true, x, y, c) -> plus(x, help(x, y, s(c))) 1114.55/291.60 if(false, x, y, c) -> 0 1114.55/291.60 lt(0, s(x)) -> true 1114.55/291.60 lt(s(x), 0) -> false 1114.55/291.60 lt(s(x), s(y)) -> lt(x, y) 1114.55/291.60 plus(x, 0) -> x 1114.55/291.60 plus(0, x) -> x 1114.55/291.60 plus(x, s(y)) -> s(plus(x, y)) 1114.55/291.60 plus(s(x), y) -> s(plus(x, y)) 1114.55/291.60 1114.55/291.60 S is empty. 1114.55/291.60 Rewrite Strategy: FULL 1114.80/291.67 EOF