304.64/291.50 WORST_CASE(Omega(n^1), ?) 304.64/291.51 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 304.64/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 304.64/291.51 304.64/291.51 304.64/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.64/291.51 304.64/291.51 (0) CpxTRS 304.64/291.51 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 304.64/291.51 (2) TRS for Loop Detection 304.64/291.51 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 304.64/291.51 (4) BEST 304.64/291.51 (5) proven lower bound 304.64/291.51 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 304.64/291.51 (7) BOUNDS(n^1, INF) 304.64/291.51 (8) TRS for Loop Detection 304.64/291.51 304.64/291.51 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (0) 304.64/291.51 Obligation: 304.64/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.64/291.51 304.64/291.51 304.64/291.51 The TRS R consists of the following rules: 304.64/291.51 304.64/291.51 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 304.64/291.51 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 304.64/291.51 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 304.64/291.51 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 304.64/291.51 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 304.64/291.51 gr(0, x) -> false 304.64/291.51 gr(s(x), 0) -> true 304.64/291.51 gr(s(x), s(y)) -> gr(x, y) 304.64/291.51 add(0, x) -> x 304.64/291.51 add(s(x), y) -> s(add(x, y)) 304.64/291.51 p(0) -> 0 304.64/291.51 p(s(x)) -> x 304.64/291.51 304.64/291.51 S is empty. 304.64/291.51 Rewrite Strategy: FULL 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 304.64/291.51 Transformed a relative TRS into a decreasing-loop problem. 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (2) 304.64/291.51 Obligation: 304.64/291.51 Analyzing the following TRS for decreasing loops: 304.64/291.51 304.64/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.64/291.51 304.64/291.51 304.64/291.51 The TRS R consists of the following rules: 304.64/291.51 304.64/291.51 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 304.64/291.51 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 304.64/291.51 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 304.64/291.51 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 304.64/291.51 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 304.64/291.51 gr(0, x) -> false 304.64/291.51 gr(s(x), 0) -> true 304.64/291.51 gr(s(x), s(y)) -> gr(x, y) 304.64/291.51 add(0, x) -> x 304.64/291.51 add(s(x), y) -> s(add(x, y)) 304.64/291.51 p(0) -> 0 304.64/291.51 p(s(x)) -> x 304.64/291.51 304.64/291.51 S is empty. 304.64/291.51 Rewrite Strategy: FULL 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (3) DecreasingLoopProof (LOWER BOUND(ID)) 304.64/291.51 The following loop(s) give(s) rise to the lower bound Omega(n^1): 304.64/291.51 304.64/291.51 The rewrite sequence 304.64/291.51 304.64/291.51 add(s(x), y) ->^+ s(add(x, y)) 304.64/291.51 304.64/291.51 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 304.64/291.51 304.64/291.51 The pumping substitution is [x / s(x)]. 304.64/291.51 304.64/291.51 The result substitution is [ ]. 304.64/291.51 304.64/291.51 304.64/291.51 304.64/291.51 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (4) 304.64/291.51 Complex Obligation (BEST) 304.64/291.51 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (5) 304.64/291.51 Obligation: 304.64/291.51 Proved the lower bound n^1 for the following obligation: 304.64/291.51 304.64/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.64/291.51 304.64/291.51 304.64/291.51 The TRS R consists of the following rules: 304.64/291.51 304.64/291.51 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 304.64/291.51 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 304.64/291.51 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 304.64/291.51 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 304.64/291.51 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 304.64/291.51 gr(0, x) -> false 304.64/291.51 gr(s(x), 0) -> true 304.64/291.51 gr(s(x), s(y)) -> gr(x, y) 304.64/291.51 add(0, x) -> x 304.64/291.51 add(s(x), y) -> s(add(x, y)) 304.64/291.51 p(0) -> 0 304.64/291.51 p(s(x)) -> x 304.64/291.51 304.64/291.51 S is empty. 304.64/291.51 Rewrite Strategy: FULL 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (6) LowerBoundPropagationProof (FINISHED) 304.64/291.51 Propagated lower bound. 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (7) 304.64/291.51 BOUNDS(n^1, INF) 304.64/291.51 304.64/291.51 ---------------------------------------- 304.64/291.51 304.64/291.51 (8) 304.64/291.51 Obligation: 304.64/291.51 Analyzing the following TRS for decreasing loops: 304.64/291.51 304.64/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 304.64/291.51 304.64/291.51 304.64/291.51 The TRS R consists of the following rules: 304.64/291.51 304.64/291.51 cond1(true, x, y) -> cond2(gr(x, 0), x, y) 304.64/291.51 cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) 304.64/291.51 cond2(false, x, y) -> cond3(gr(y, 0), x, y) 304.64/291.51 cond3(true, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) 304.64/291.51 cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, y) 304.64/291.51 gr(0, x) -> false 304.64/291.51 gr(s(x), 0) -> true 304.64/291.51 gr(s(x), s(y)) -> gr(x, y) 304.64/291.51 add(0, x) -> x 304.64/291.51 add(s(x), y) -> s(add(x, y)) 304.64/291.51 p(0) -> 0 304.64/291.51 p(s(x)) -> x 304.64/291.51 304.64/291.51 S is empty. 304.64/291.51 Rewrite Strategy: FULL 304.73/291.54 EOF