1104.28/291.72 WORST_CASE(Omega(n^1), ?) 1104.28/291.74 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1104.28/291.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1104.28/291.74 1104.28/291.74 1104.28/291.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.28/291.74 1104.28/291.74 (0) CpxTRS 1104.28/291.74 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1104.28/291.74 (2) TRS for Loop Detection 1104.28/291.74 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1104.28/291.74 (4) BEST 1104.28/291.74 (5) proven lower bound 1104.28/291.74 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1104.28/291.74 (7) BOUNDS(n^1, INF) 1104.28/291.74 (8) TRS for Loop Detection 1104.28/291.74 1104.28/291.74 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (0) 1104.28/291.74 Obligation: 1104.28/291.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.28/291.74 1104.28/291.74 1104.28/291.74 The TRS R consists of the following rules: 1104.28/291.74 1104.28/291.74 times(x, y) -> sum(generate(x, y)) 1104.28/291.74 generate(x, y) -> gen(x, y, 0) 1104.28/291.74 gen(x, y, z) -> if(ge(z, x), x, y, z) 1104.28/291.74 if(true, x, y, z) -> nil 1104.28/291.74 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1104.28/291.74 sum(nil) -> 0 1104.28/291.74 sum(cons(0, xs)) -> sum(xs) 1104.28/291.74 sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) 1104.28/291.74 ge(x, 0) -> true 1104.28/291.74 ge(0, s(y)) -> false 1104.28/291.74 ge(s(x), s(y)) -> ge(x, y) 1104.28/291.74 1104.28/291.74 S is empty. 1104.28/291.74 Rewrite Strategy: FULL 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1104.28/291.74 Transformed a relative TRS into a decreasing-loop problem. 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (2) 1104.28/291.74 Obligation: 1104.28/291.74 Analyzing the following TRS for decreasing loops: 1104.28/291.74 1104.28/291.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.28/291.74 1104.28/291.74 1104.28/291.74 The TRS R consists of the following rules: 1104.28/291.74 1104.28/291.74 times(x, y) -> sum(generate(x, y)) 1104.28/291.74 generate(x, y) -> gen(x, y, 0) 1104.28/291.74 gen(x, y, z) -> if(ge(z, x), x, y, z) 1104.28/291.74 if(true, x, y, z) -> nil 1104.28/291.74 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1104.28/291.74 sum(nil) -> 0 1104.28/291.74 sum(cons(0, xs)) -> sum(xs) 1104.28/291.74 sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) 1104.28/291.74 ge(x, 0) -> true 1104.28/291.74 ge(0, s(y)) -> false 1104.28/291.74 ge(s(x), s(y)) -> ge(x, y) 1104.28/291.74 1104.28/291.74 S is empty. 1104.28/291.74 Rewrite Strategy: FULL 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1104.28/291.74 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1104.28/291.74 1104.28/291.74 The rewrite sequence 1104.28/291.74 1104.28/291.74 sum(cons(0, xs)) ->^+ sum(xs) 1104.28/291.74 1104.28/291.74 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1104.28/291.74 1104.28/291.74 The pumping substitution is [xs / cons(0, xs)]. 1104.28/291.74 1104.28/291.74 The result substitution is [ ]. 1104.28/291.74 1104.28/291.74 1104.28/291.74 1104.28/291.74 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (4) 1104.28/291.74 Complex Obligation (BEST) 1104.28/291.74 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (5) 1104.28/291.74 Obligation: 1104.28/291.74 Proved the lower bound n^1 for the following obligation: 1104.28/291.74 1104.28/291.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.28/291.74 1104.28/291.74 1104.28/291.74 The TRS R consists of the following rules: 1104.28/291.74 1104.28/291.74 times(x, y) -> sum(generate(x, y)) 1104.28/291.74 generate(x, y) -> gen(x, y, 0) 1104.28/291.74 gen(x, y, z) -> if(ge(z, x), x, y, z) 1104.28/291.74 if(true, x, y, z) -> nil 1104.28/291.74 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1104.28/291.74 sum(nil) -> 0 1104.28/291.74 sum(cons(0, xs)) -> sum(xs) 1104.28/291.74 sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) 1104.28/291.74 ge(x, 0) -> true 1104.28/291.74 ge(0, s(y)) -> false 1104.28/291.74 ge(s(x), s(y)) -> ge(x, y) 1104.28/291.74 1104.28/291.74 S is empty. 1104.28/291.74 Rewrite Strategy: FULL 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (6) LowerBoundPropagationProof (FINISHED) 1104.28/291.74 Propagated lower bound. 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (7) 1104.28/291.74 BOUNDS(n^1, INF) 1104.28/291.74 1104.28/291.74 ---------------------------------------- 1104.28/291.74 1104.28/291.74 (8) 1104.28/291.74 Obligation: 1104.28/291.74 Analyzing the following TRS for decreasing loops: 1104.28/291.74 1104.28/291.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.28/291.74 1104.28/291.74 1104.28/291.74 The TRS R consists of the following rules: 1104.28/291.74 1104.28/291.74 times(x, y) -> sum(generate(x, y)) 1104.28/291.74 generate(x, y) -> gen(x, y, 0) 1104.28/291.74 gen(x, y, z) -> if(ge(z, x), x, y, z) 1104.28/291.74 if(true, x, y, z) -> nil 1104.28/291.74 if(false, x, y, z) -> cons(y, gen(x, y, s(z))) 1104.28/291.74 sum(nil) -> 0 1104.28/291.74 sum(cons(0, xs)) -> sum(xs) 1104.28/291.74 sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) 1104.28/291.74 ge(x, 0) -> true 1104.28/291.74 ge(0, s(y)) -> false 1104.28/291.74 ge(s(x), s(y)) -> ge(x, y) 1104.28/291.74 1104.28/291.74 S is empty. 1104.28/291.74 Rewrite Strategy: FULL 1104.69/291.81 EOF