1090.35/291.54 WORST_CASE(Omega(n^1), ?) 1090.35/291.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1090.35/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1090.35/291.57 1090.35/291.57 1090.35/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1090.35/291.57 1090.35/291.57 (0) CpxTRS 1090.35/291.57 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1090.35/291.57 (2) TRS for Loop Detection 1090.35/291.57 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1090.35/291.57 (4) BEST 1090.35/291.57 (5) proven lower bound 1090.35/291.57 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1090.35/291.57 (7) BOUNDS(n^1, INF) 1090.35/291.57 (8) TRS for Loop Detection 1090.35/291.57 1090.35/291.57 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (0) 1090.35/291.57 Obligation: 1090.35/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1090.35/291.57 1090.35/291.57 1090.35/291.57 The TRS R consists of the following rules: 1090.35/291.57 1090.35/291.57 top1(free(x), y) -> top2(check(new(x)), y) 1090.35/291.57 top1(free(x), y) -> top2(new(x), check(y)) 1090.35/291.57 top1(free(x), y) -> top2(check(x), new(y)) 1090.35/291.57 top1(free(x), y) -> top2(x, check(new(y))) 1090.35/291.57 top2(x, free(y)) -> top1(check(new(x)), y) 1090.35/291.57 top2(x, free(y)) -> top1(new(x), check(y)) 1090.35/291.57 top2(x, free(y)) -> top1(check(x), new(y)) 1090.35/291.57 top2(x, free(y)) -> top1(x, check(new(y))) 1090.35/291.57 new(free(x)) -> free(new(x)) 1090.35/291.57 old(free(x)) -> free(old(x)) 1090.35/291.57 new(serve) -> free(serve) 1090.35/291.57 old(serve) -> free(serve) 1090.35/291.57 check(free(x)) -> free(check(x)) 1090.35/291.57 check(new(x)) -> new(check(x)) 1090.35/291.57 check(old(x)) -> old(check(x)) 1090.35/291.57 check(old(x)) -> old(x) 1090.35/291.57 1090.35/291.57 S is empty. 1090.35/291.57 Rewrite Strategy: FULL 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1090.35/291.57 Transformed a relative TRS into a decreasing-loop problem. 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (2) 1090.35/291.57 Obligation: 1090.35/291.57 Analyzing the following TRS for decreasing loops: 1090.35/291.57 1090.35/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1090.35/291.57 1090.35/291.57 1090.35/291.57 The TRS R consists of the following rules: 1090.35/291.57 1090.35/291.57 top1(free(x), y) -> top2(check(new(x)), y) 1090.35/291.57 top1(free(x), y) -> top2(new(x), check(y)) 1090.35/291.57 top1(free(x), y) -> top2(check(x), new(y)) 1090.35/291.57 top1(free(x), y) -> top2(x, check(new(y))) 1090.35/291.57 top2(x, free(y)) -> top1(check(new(x)), y) 1090.35/291.57 top2(x, free(y)) -> top1(new(x), check(y)) 1090.35/291.57 top2(x, free(y)) -> top1(check(x), new(y)) 1090.35/291.57 top2(x, free(y)) -> top1(x, check(new(y))) 1090.35/291.57 new(free(x)) -> free(new(x)) 1090.35/291.57 old(free(x)) -> free(old(x)) 1090.35/291.57 new(serve) -> free(serve) 1090.35/291.57 old(serve) -> free(serve) 1090.35/291.57 check(free(x)) -> free(check(x)) 1090.35/291.57 check(new(x)) -> new(check(x)) 1090.35/291.57 check(old(x)) -> old(check(x)) 1090.35/291.57 check(old(x)) -> old(x) 1090.35/291.57 1090.35/291.57 S is empty. 1090.35/291.57 Rewrite Strategy: FULL 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1090.35/291.57 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1090.35/291.57 1090.35/291.57 The rewrite sequence 1090.35/291.57 1090.35/291.57 check(free(x)) ->^+ free(check(x)) 1090.35/291.57 1090.35/291.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1090.35/291.57 1090.35/291.57 The pumping substitution is [x / free(x)]. 1090.35/291.57 1090.35/291.57 The result substitution is [ ]. 1090.35/291.57 1090.35/291.57 1090.35/291.57 1090.35/291.57 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (4) 1090.35/291.57 Complex Obligation (BEST) 1090.35/291.57 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (5) 1090.35/291.57 Obligation: 1090.35/291.57 Proved the lower bound n^1 for the following obligation: 1090.35/291.57 1090.35/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1090.35/291.57 1090.35/291.57 1090.35/291.57 The TRS R consists of the following rules: 1090.35/291.57 1090.35/291.57 top1(free(x), y) -> top2(check(new(x)), y) 1090.35/291.57 top1(free(x), y) -> top2(new(x), check(y)) 1090.35/291.57 top1(free(x), y) -> top2(check(x), new(y)) 1090.35/291.57 top1(free(x), y) -> top2(x, check(new(y))) 1090.35/291.57 top2(x, free(y)) -> top1(check(new(x)), y) 1090.35/291.57 top2(x, free(y)) -> top1(new(x), check(y)) 1090.35/291.57 top2(x, free(y)) -> top1(check(x), new(y)) 1090.35/291.57 top2(x, free(y)) -> top1(x, check(new(y))) 1090.35/291.57 new(free(x)) -> free(new(x)) 1090.35/291.57 old(free(x)) -> free(old(x)) 1090.35/291.57 new(serve) -> free(serve) 1090.35/291.57 old(serve) -> free(serve) 1090.35/291.57 check(free(x)) -> free(check(x)) 1090.35/291.57 check(new(x)) -> new(check(x)) 1090.35/291.57 check(old(x)) -> old(check(x)) 1090.35/291.57 check(old(x)) -> old(x) 1090.35/291.57 1090.35/291.57 S is empty. 1090.35/291.57 Rewrite Strategy: FULL 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (6) LowerBoundPropagationProof (FINISHED) 1090.35/291.57 Propagated lower bound. 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (7) 1090.35/291.57 BOUNDS(n^1, INF) 1090.35/291.57 1090.35/291.57 ---------------------------------------- 1090.35/291.57 1090.35/291.57 (8) 1090.35/291.57 Obligation: 1090.35/291.57 Analyzing the following TRS for decreasing loops: 1090.35/291.57 1090.35/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1090.35/291.57 1090.35/291.57 1090.35/291.57 The TRS R consists of the following rules: 1090.35/291.57 1090.35/291.57 top1(free(x), y) -> top2(check(new(x)), y) 1090.35/291.57 top1(free(x), y) -> top2(new(x), check(y)) 1090.35/291.57 top1(free(x), y) -> top2(check(x), new(y)) 1090.35/291.57 top1(free(x), y) -> top2(x, check(new(y))) 1090.35/291.57 top2(x, free(y)) -> top1(check(new(x)), y) 1090.35/291.57 top2(x, free(y)) -> top1(new(x), check(y)) 1090.35/291.57 top2(x, free(y)) -> top1(check(x), new(y)) 1090.35/291.57 top2(x, free(y)) -> top1(x, check(new(y))) 1090.35/291.57 new(free(x)) -> free(new(x)) 1090.35/291.57 old(free(x)) -> free(old(x)) 1090.35/291.57 new(serve) -> free(serve) 1090.35/291.57 old(serve) -> free(serve) 1090.35/291.57 check(free(x)) -> free(check(x)) 1090.35/291.57 check(new(x)) -> new(check(x)) 1090.35/291.57 check(old(x)) -> old(check(x)) 1090.35/291.57 check(old(x)) -> old(x) 1090.35/291.57 1090.35/291.57 S is empty. 1090.35/291.57 Rewrite Strategy: FULL 1090.74/291.74 EOF