305.70/291.49 WORST_CASE(Omega(n^1), ?) 305.70/291.50 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 305.70/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.70/291.50 305.70/291.50 305.70/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.70/291.50 305.70/291.50 (0) CpxTRS 305.70/291.50 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 305.70/291.50 (2) TRS for Loop Detection 305.70/291.50 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 305.70/291.50 (4) BEST 305.70/291.50 (5) proven lower bound 305.70/291.50 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 305.70/291.50 (7) BOUNDS(n^1, INF) 305.70/291.50 (8) TRS for Loop Detection 305.70/291.50 305.70/291.50 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (0) 305.70/291.50 Obligation: 305.70/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.70/291.50 305.70/291.50 305.70/291.50 The TRS R consists of the following rules: 305.70/291.50 305.70/291.50 le(0, y) -> true 305.70/291.50 le(s(x), 0) -> false 305.70/291.50 le(s(x), s(y)) -> le(x, y) 305.70/291.50 minus(x, 0) -> x 305.70/291.50 minus(0, x) -> 0 305.70/291.50 minus(s(x), s(y)) -> minus(x, y) 305.70/291.50 gcd(0, y) -> y 305.70/291.50 gcd(s(x), 0) -> s(x) 305.70/291.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 305.70/291.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 305.70/291.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 305.70/291.50 305.70/291.50 S is empty. 305.70/291.50 Rewrite Strategy: FULL 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 305.70/291.50 Transformed a relative TRS into a decreasing-loop problem. 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (2) 305.70/291.50 Obligation: 305.70/291.50 Analyzing the following TRS for decreasing loops: 305.70/291.50 305.70/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.70/291.50 305.70/291.50 305.70/291.50 The TRS R consists of the following rules: 305.70/291.50 305.70/291.50 le(0, y) -> true 305.70/291.50 le(s(x), 0) -> false 305.70/291.50 le(s(x), s(y)) -> le(x, y) 305.70/291.50 minus(x, 0) -> x 305.70/291.50 minus(0, x) -> 0 305.70/291.50 minus(s(x), s(y)) -> minus(x, y) 305.70/291.50 gcd(0, y) -> y 305.70/291.50 gcd(s(x), 0) -> s(x) 305.70/291.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 305.70/291.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 305.70/291.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 305.70/291.50 305.70/291.50 S is empty. 305.70/291.50 Rewrite Strategy: FULL 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (3) DecreasingLoopProof (LOWER BOUND(ID)) 305.70/291.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 305.70/291.50 305.70/291.50 The rewrite sequence 305.70/291.50 305.70/291.50 minus(s(x), s(y)) ->^+ minus(x, y) 305.70/291.50 305.70/291.50 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 305.70/291.50 305.70/291.50 The pumping substitution is [x / s(x), y / s(y)]. 305.70/291.50 305.70/291.50 The result substitution is [ ]. 305.70/291.50 305.70/291.50 305.70/291.50 305.70/291.50 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (4) 305.70/291.50 Complex Obligation (BEST) 305.70/291.50 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (5) 305.70/291.50 Obligation: 305.70/291.50 Proved the lower bound n^1 for the following obligation: 305.70/291.50 305.70/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.70/291.50 305.70/291.50 305.70/291.50 The TRS R consists of the following rules: 305.70/291.50 305.70/291.50 le(0, y) -> true 305.70/291.50 le(s(x), 0) -> false 305.70/291.50 le(s(x), s(y)) -> le(x, y) 305.70/291.50 minus(x, 0) -> x 305.70/291.50 minus(0, x) -> 0 305.70/291.50 minus(s(x), s(y)) -> minus(x, y) 305.70/291.50 gcd(0, y) -> y 305.70/291.50 gcd(s(x), 0) -> s(x) 305.70/291.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 305.70/291.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 305.70/291.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 305.70/291.50 305.70/291.50 S is empty. 305.70/291.50 Rewrite Strategy: FULL 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (6) LowerBoundPropagationProof (FINISHED) 305.70/291.50 Propagated lower bound. 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (7) 305.70/291.50 BOUNDS(n^1, INF) 305.70/291.50 305.70/291.50 ---------------------------------------- 305.70/291.50 305.70/291.50 (8) 305.70/291.50 Obligation: 305.70/291.50 Analyzing the following TRS for decreasing loops: 305.70/291.50 305.70/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.70/291.50 305.70/291.50 305.70/291.50 The TRS R consists of the following rules: 305.70/291.50 305.70/291.50 le(0, y) -> true 305.70/291.50 le(s(x), 0) -> false 305.70/291.50 le(s(x), s(y)) -> le(x, y) 305.70/291.50 minus(x, 0) -> x 305.70/291.50 minus(0, x) -> 0 305.70/291.50 minus(s(x), s(y)) -> minus(x, y) 305.70/291.50 gcd(0, y) -> y 305.70/291.50 gcd(s(x), 0) -> s(x) 305.70/291.50 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 305.70/291.50 if_gcd(true, x, y) -> gcd(minus(x, y), y) 305.70/291.50 if_gcd(false, x, y) -> gcd(minus(y, x), x) 305.70/291.50 305.70/291.50 S is empty. 305.70/291.50 Rewrite Strategy: FULL 305.77/291.52 EOF