301.77/291.54 WORST_CASE(Omega(n^1), ?) 301.77/291.55 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 301.77/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 301.77/291.55 301.77/291.55 301.77/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 301.77/291.55 301.77/291.55 (0) CpxTRS 301.77/291.55 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 301.77/291.55 (2) TRS for Loop Detection 301.77/291.55 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 301.77/291.55 (4) BEST 301.77/291.55 (5) proven lower bound 301.77/291.55 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 301.77/291.55 (7) BOUNDS(n^1, INF) 301.77/291.55 (8) TRS for Loop Detection 301.77/291.55 301.77/291.55 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (0) 301.77/291.55 Obligation: 301.77/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 301.77/291.55 301.77/291.55 301.77/291.55 The TRS R consists of the following rules: 301.77/291.55 301.77/291.55 minus(s(x), y) -> if(gt(s(x), y), x, y) 301.77/291.55 if(true, x, y) -> s(minus(x, y)) 301.77/291.55 if(false, x, y) -> 0 301.77/291.55 gcd(x, y) -> if1(ge(x, y), x, y) 301.77/291.55 if1(true, x, y) -> if2(gt(y, 0), x, y) 301.77/291.55 if1(false, x, y) -> if3(gt(x, 0), x, y) 301.77/291.55 if2(true, x, y) -> gcd(minus(x, y), y) 301.77/291.55 if2(false, x, y) -> x 301.77/291.55 if3(true, x, y) -> gcd(x, minus(y, x)) 301.77/291.55 if3(false, x, y) -> y 301.77/291.55 gt(0, y) -> false 301.77/291.55 gt(s(x), 0) -> true 301.77/291.55 gt(s(x), s(y)) -> gt(x, y) 301.77/291.55 ge(x, 0) -> true 301.77/291.55 ge(0, s(x)) -> false 301.77/291.55 ge(s(x), s(y)) -> ge(x, y) 301.77/291.55 301.77/291.55 S is empty. 301.77/291.55 Rewrite Strategy: FULL 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 301.77/291.55 Transformed a relative TRS into a decreasing-loop problem. 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (2) 301.77/291.55 Obligation: 301.77/291.55 Analyzing the following TRS for decreasing loops: 301.77/291.55 301.77/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 301.77/291.55 301.77/291.55 301.77/291.55 The TRS R consists of the following rules: 301.77/291.55 301.77/291.55 minus(s(x), y) -> if(gt(s(x), y), x, y) 301.77/291.55 if(true, x, y) -> s(minus(x, y)) 301.77/291.55 if(false, x, y) -> 0 301.77/291.55 gcd(x, y) -> if1(ge(x, y), x, y) 301.77/291.55 if1(true, x, y) -> if2(gt(y, 0), x, y) 301.77/291.55 if1(false, x, y) -> if3(gt(x, 0), x, y) 301.77/291.55 if2(true, x, y) -> gcd(minus(x, y), y) 301.77/291.55 if2(false, x, y) -> x 301.77/291.55 if3(true, x, y) -> gcd(x, minus(y, x)) 301.77/291.55 if3(false, x, y) -> y 301.77/291.55 gt(0, y) -> false 301.77/291.55 gt(s(x), 0) -> true 301.77/291.55 gt(s(x), s(y)) -> gt(x, y) 301.77/291.55 ge(x, 0) -> true 301.77/291.55 ge(0, s(x)) -> false 301.77/291.55 ge(s(x), s(y)) -> ge(x, y) 301.77/291.55 301.77/291.55 S is empty. 301.77/291.55 Rewrite Strategy: FULL 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (3) DecreasingLoopProof (LOWER BOUND(ID)) 301.77/291.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 301.77/291.55 301.77/291.55 The rewrite sequence 301.77/291.55 301.77/291.55 gt(s(x), s(y)) ->^+ gt(x, y) 301.77/291.55 301.77/291.55 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 301.77/291.55 301.77/291.55 The pumping substitution is [x / s(x), y / s(y)]. 301.77/291.55 301.77/291.55 The result substitution is [ ]. 301.77/291.55 301.77/291.55 301.77/291.55 301.77/291.55 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (4) 301.77/291.55 Complex Obligation (BEST) 301.77/291.55 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (5) 301.77/291.55 Obligation: 301.77/291.55 Proved the lower bound n^1 for the following obligation: 301.77/291.55 301.77/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 301.77/291.55 301.77/291.55 301.77/291.55 The TRS R consists of the following rules: 301.77/291.55 301.77/291.55 minus(s(x), y) -> if(gt(s(x), y), x, y) 301.77/291.55 if(true, x, y) -> s(minus(x, y)) 301.77/291.55 if(false, x, y) -> 0 301.77/291.55 gcd(x, y) -> if1(ge(x, y), x, y) 301.77/291.55 if1(true, x, y) -> if2(gt(y, 0), x, y) 301.77/291.55 if1(false, x, y) -> if3(gt(x, 0), x, y) 301.77/291.55 if2(true, x, y) -> gcd(minus(x, y), y) 301.77/291.55 if2(false, x, y) -> x 301.77/291.55 if3(true, x, y) -> gcd(x, minus(y, x)) 301.77/291.55 if3(false, x, y) -> y 301.77/291.55 gt(0, y) -> false 301.77/291.55 gt(s(x), 0) -> true 301.77/291.55 gt(s(x), s(y)) -> gt(x, y) 301.77/291.55 ge(x, 0) -> true 301.77/291.55 ge(0, s(x)) -> false 301.77/291.55 ge(s(x), s(y)) -> ge(x, y) 301.77/291.55 301.77/291.55 S is empty. 301.77/291.55 Rewrite Strategy: FULL 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (6) LowerBoundPropagationProof (FINISHED) 301.77/291.55 Propagated lower bound. 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (7) 301.77/291.55 BOUNDS(n^1, INF) 301.77/291.55 301.77/291.55 ---------------------------------------- 301.77/291.55 301.77/291.55 (8) 301.77/291.55 Obligation: 301.77/291.55 Analyzing the following TRS for decreasing loops: 301.77/291.55 301.77/291.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 301.77/291.55 301.77/291.55 301.77/291.55 The TRS R consists of the following rules: 301.77/291.55 301.77/291.55 minus(s(x), y) -> if(gt(s(x), y), x, y) 301.77/291.55 if(true, x, y) -> s(minus(x, y)) 301.77/291.55 if(false, x, y) -> 0 301.77/291.55 gcd(x, y) -> if1(ge(x, y), x, y) 301.77/291.55 if1(true, x, y) -> if2(gt(y, 0), x, y) 301.77/291.55 if1(false, x, y) -> if3(gt(x, 0), x, y) 301.77/291.55 if2(true, x, y) -> gcd(minus(x, y), y) 301.77/291.55 if2(false, x, y) -> x 301.77/291.55 if3(true, x, y) -> gcd(x, minus(y, x)) 301.77/291.55 if3(false, x, y) -> y 301.77/291.55 gt(0, y) -> false 301.77/291.55 gt(s(x), 0) -> true 301.77/291.55 gt(s(x), s(y)) -> gt(x, y) 301.77/291.55 ge(x, 0) -> true 301.77/291.55 ge(0, s(x)) -> false 301.77/291.55 ge(s(x), s(y)) -> ge(x, y) 301.77/291.55 301.77/291.55 S is empty. 301.77/291.55 Rewrite Strategy: FULL 301.77/291.59 EOF