1111.85/291.49 WORST_CASE(Omega(n^1), ?) 1112.08/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1112.08/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1112.08/291.52 1112.08/291.52 1112.08/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1112.08/291.52 1112.08/291.52 (0) CpxTRS 1112.08/291.52 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1112.08/291.52 (2) TRS for Loop Detection 1112.08/291.52 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1112.08/291.52 (4) BEST 1112.08/291.52 (5) proven lower bound 1112.08/291.52 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1112.08/291.52 (7) BOUNDS(n^1, INF) 1112.08/291.52 (8) TRS for Loop Detection 1112.08/291.52 1112.08/291.52 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (0) 1112.08/291.52 Obligation: 1112.08/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1112.08/291.52 1112.08/291.52 1112.08/291.52 The TRS R consists of the following rules: 1112.08/291.52 1112.08/291.52 le(0, y) -> true 1112.08/291.52 le(s(x), 0) -> false 1112.08/291.52 le(s(x), s(y)) -> le(x, y) 1112.08/291.52 minus(x, 0) -> x 1112.08/291.52 minus(0, s(y)) -> 0 1112.08/291.52 minus(s(x), s(y)) -> minus(x, y) 1112.08/291.52 plus(x, 0) -> x 1112.08/291.52 plus(x, s(y)) -> s(plus(x, y)) 1112.08/291.52 mod(s(x), 0) -> 0 1112.08/291.52 mod(x, s(y)) -> help(x, s(y), 0) 1112.08/291.52 help(x, s(y), c) -> if(le(c, x), x, s(y), c) 1112.08/291.52 if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) 1112.08/291.52 if(false, x, s(y), c) -> minus(x, minus(c, s(y))) 1112.08/291.52 1112.08/291.52 S is empty. 1112.08/291.52 Rewrite Strategy: INNERMOST 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1112.08/291.52 Transformed a relative TRS into a decreasing-loop problem. 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (2) 1112.08/291.52 Obligation: 1112.08/291.52 Analyzing the following TRS for decreasing loops: 1112.08/291.52 1112.08/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1112.08/291.52 1112.08/291.52 1112.08/291.52 The TRS R consists of the following rules: 1112.08/291.52 1112.08/291.52 le(0, y) -> true 1112.08/291.52 le(s(x), 0) -> false 1112.08/291.52 le(s(x), s(y)) -> le(x, y) 1112.08/291.52 minus(x, 0) -> x 1112.08/291.52 minus(0, s(y)) -> 0 1112.08/291.52 minus(s(x), s(y)) -> minus(x, y) 1112.08/291.52 plus(x, 0) -> x 1112.08/291.52 plus(x, s(y)) -> s(plus(x, y)) 1112.08/291.52 mod(s(x), 0) -> 0 1112.08/291.52 mod(x, s(y)) -> help(x, s(y), 0) 1112.08/291.52 help(x, s(y), c) -> if(le(c, x), x, s(y), c) 1112.08/291.52 if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) 1112.08/291.52 if(false, x, s(y), c) -> minus(x, minus(c, s(y))) 1112.08/291.52 1112.08/291.52 S is empty. 1112.08/291.52 Rewrite Strategy: INNERMOST 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1112.08/291.52 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1112.08/291.52 1112.08/291.52 The rewrite sequence 1112.08/291.52 1112.08/291.52 le(s(x), s(y)) ->^+ le(x, y) 1112.08/291.52 1112.08/291.52 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1112.08/291.52 1112.08/291.52 The pumping substitution is [x / s(x), y / s(y)]. 1112.08/291.52 1112.08/291.52 The result substitution is [ ]. 1112.08/291.52 1112.08/291.52 1112.08/291.52 1112.08/291.52 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (4) 1112.08/291.52 Complex Obligation (BEST) 1112.08/291.52 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (5) 1112.08/291.52 Obligation: 1112.08/291.52 Proved the lower bound n^1 for the following obligation: 1112.08/291.52 1112.08/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1112.08/291.52 1112.08/291.52 1112.08/291.52 The TRS R consists of the following rules: 1112.08/291.52 1112.08/291.52 le(0, y) -> true 1112.08/291.52 le(s(x), 0) -> false 1112.08/291.52 le(s(x), s(y)) -> le(x, y) 1112.08/291.52 minus(x, 0) -> x 1112.08/291.52 minus(0, s(y)) -> 0 1112.08/291.52 minus(s(x), s(y)) -> minus(x, y) 1112.08/291.52 plus(x, 0) -> x 1112.08/291.52 plus(x, s(y)) -> s(plus(x, y)) 1112.08/291.52 mod(s(x), 0) -> 0 1112.08/291.52 mod(x, s(y)) -> help(x, s(y), 0) 1112.08/291.52 help(x, s(y), c) -> if(le(c, x), x, s(y), c) 1112.08/291.52 if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) 1112.08/291.52 if(false, x, s(y), c) -> minus(x, minus(c, s(y))) 1112.08/291.52 1112.08/291.52 S is empty. 1112.08/291.52 Rewrite Strategy: INNERMOST 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (6) LowerBoundPropagationProof (FINISHED) 1112.08/291.52 Propagated lower bound. 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (7) 1112.08/291.52 BOUNDS(n^1, INF) 1112.08/291.52 1112.08/291.52 ---------------------------------------- 1112.08/291.52 1112.08/291.52 (8) 1112.08/291.52 Obligation: 1112.08/291.52 Analyzing the following TRS for decreasing loops: 1112.08/291.52 1112.08/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1112.08/291.52 1112.08/291.52 1112.08/291.52 The TRS R consists of the following rules: 1112.08/291.52 1112.08/291.52 le(0, y) -> true 1112.08/291.52 le(s(x), 0) -> false 1112.08/291.52 le(s(x), s(y)) -> le(x, y) 1112.08/291.52 minus(x, 0) -> x 1112.08/291.52 minus(0, s(y)) -> 0 1112.08/291.52 minus(s(x), s(y)) -> minus(x, y) 1112.08/291.52 plus(x, 0) -> x 1112.08/291.52 plus(x, s(y)) -> s(plus(x, y)) 1112.08/291.52 mod(s(x), 0) -> 0 1112.08/291.52 mod(x, s(y)) -> help(x, s(y), 0) 1112.08/291.52 help(x, s(y), c) -> if(le(c, x), x, s(y), c) 1112.08/291.52 if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) 1112.08/291.52 if(false, x, s(y), c) -> minus(x, minus(c, s(y))) 1112.08/291.52 1112.08/291.52 S is empty. 1112.08/291.52 Rewrite Strategy: INNERMOST 1112.17/291.59 EOF