310.55/291.55 WORST_CASE(Omega(n^1), ?) 310.70/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 310.70/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 310.70/291.58 310.70/291.58 310.70/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.70/291.58 310.70/291.58 (0) CpxTRS 310.70/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 310.70/291.58 (2) TRS for Loop Detection 310.70/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 310.70/291.58 (4) BEST 310.70/291.58 (5) proven lower bound 310.70/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 310.70/291.58 (7) BOUNDS(n^1, INF) 310.70/291.58 (8) TRS for Loop Detection 310.70/291.58 310.70/291.58 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (0) 310.70/291.58 Obligation: 310.70/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.70/291.58 310.70/291.58 310.70/291.58 The TRS R consists of the following rules: 310.70/291.58 310.70/291.58 minus(X, s(Y)) -> pred(minus(X, Y)) 310.70/291.58 minus(X, 0) -> X 310.70/291.58 pred(s(X)) -> X 310.70/291.58 le(s(X), s(Y)) -> le(X, Y) 310.70/291.58 le(s(X), 0) -> false 310.70/291.58 le(0, Y) -> true 310.70/291.58 gcd(0, Y) -> 0 310.70/291.58 gcd(s(X), 0) -> s(X) 310.70/291.58 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 310.70/291.58 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 310.70/291.58 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 310.70/291.58 310.70/291.58 S is empty. 310.70/291.58 Rewrite Strategy: FULL 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 310.70/291.58 Transformed a relative TRS into a decreasing-loop problem. 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (2) 310.70/291.58 Obligation: 310.70/291.58 Analyzing the following TRS for decreasing loops: 310.70/291.58 310.70/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.70/291.58 310.70/291.58 310.70/291.58 The TRS R consists of the following rules: 310.70/291.58 310.70/291.58 minus(X, s(Y)) -> pred(minus(X, Y)) 310.70/291.58 minus(X, 0) -> X 310.70/291.58 pred(s(X)) -> X 310.70/291.58 le(s(X), s(Y)) -> le(X, Y) 310.70/291.58 le(s(X), 0) -> false 310.70/291.58 le(0, Y) -> true 310.70/291.58 gcd(0, Y) -> 0 310.70/291.58 gcd(s(X), 0) -> s(X) 310.70/291.58 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 310.70/291.58 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 310.70/291.58 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 310.70/291.58 310.70/291.58 S is empty. 310.70/291.58 Rewrite Strategy: FULL 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 310.70/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 310.70/291.58 310.70/291.58 The rewrite sequence 310.70/291.58 310.70/291.58 minus(X, s(Y)) ->^+ pred(minus(X, Y)) 310.70/291.58 310.70/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 310.70/291.58 310.70/291.58 The pumping substitution is [Y / s(Y)]. 310.70/291.58 310.70/291.58 The result substitution is [ ]. 310.70/291.58 310.70/291.58 310.70/291.58 310.70/291.58 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (4) 310.70/291.58 Complex Obligation (BEST) 310.70/291.58 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (5) 310.70/291.58 Obligation: 310.70/291.58 Proved the lower bound n^1 for the following obligation: 310.70/291.58 310.70/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.70/291.58 310.70/291.58 310.70/291.58 The TRS R consists of the following rules: 310.70/291.58 310.70/291.58 minus(X, s(Y)) -> pred(minus(X, Y)) 310.70/291.58 minus(X, 0) -> X 310.70/291.58 pred(s(X)) -> X 310.70/291.58 le(s(X), s(Y)) -> le(X, Y) 310.70/291.58 le(s(X), 0) -> false 310.70/291.58 le(0, Y) -> true 310.70/291.58 gcd(0, Y) -> 0 310.70/291.58 gcd(s(X), 0) -> s(X) 310.70/291.58 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 310.70/291.58 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 310.70/291.58 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 310.70/291.58 310.70/291.58 S is empty. 310.70/291.58 Rewrite Strategy: FULL 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (6) LowerBoundPropagationProof (FINISHED) 310.70/291.58 Propagated lower bound. 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (7) 310.70/291.58 BOUNDS(n^1, INF) 310.70/291.58 310.70/291.58 ---------------------------------------- 310.70/291.58 310.70/291.58 (8) 310.70/291.58 Obligation: 310.70/291.58 Analyzing the following TRS for decreasing loops: 310.70/291.58 310.70/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 310.70/291.58 310.70/291.58 310.70/291.58 The TRS R consists of the following rules: 310.70/291.58 310.70/291.58 minus(X, s(Y)) -> pred(minus(X, Y)) 310.70/291.58 minus(X, 0) -> X 310.70/291.58 pred(s(X)) -> X 310.70/291.58 le(s(X), s(Y)) -> le(X, Y) 310.70/291.58 le(s(X), 0) -> false 310.70/291.58 le(0, Y) -> true 310.70/291.58 gcd(0, Y) -> 0 310.70/291.58 gcd(s(X), 0) -> s(X) 310.70/291.58 gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y)) 310.70/291.58 if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y)) 310.70/291.58 if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X)) 310.70/291.58 310.70/291.58 S is empty. 310.70/291.58 Rewrite Strategy: FULL 310.70/291.61 EOF