305.45/291.48 WORST_CASE(Omega(n^1), ?) 305.45/291.48 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 305.45/291.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.45/291.48 305.45/291.48 305.45/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.45/291.48 305.45/291.48 (0) CpxTRS 305.45/291.48 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 305.45/291.48 (2) TRS for Loop Detection 305.45/291.48 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 305.45/291.48 (4) BEST 305.45/291.48 (5) proven lower bound 305.45/291.48 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 305.45/291.48 (7) BOUNDS(n^1, INF) 305.45/291.48 (8) TRS for Loop Detection 305.45/291.48 305.45/291.48 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (0) 305.45/291.48 Obligation: 305.45/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.45/291.48 305.45/291.48 305.45/291.48 The TRS R consists of the following rules: 305.45/291.48 305.45/291.48 cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) 305.45/291.48 cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z) 305.45/291.48 cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) 305.45/291.48 cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z) 305.45/291.48 cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z) 305.45/291.48 gr(0, x) -> false 305.45/291.48 gr(s(x), 0) -> true 305.45/291.48 gr(s(x), s(y)) -> gr(x, y) 305.45/291.48 add(0, x) -> x 305.45/291.48 add(s(x), y) -> s(add(x, y)) 305.45/291.48 p(0) -> 0 305.45/291.48 p(s(x)) -> x 305.45/291.48 305.45/291.48 S is empty. 305.45/291.48 Rewrite Strategy: FULL 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 305.45/291.48 Transformed a relative TRS into a decreasing-loop problem. 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (2) 305.45/291.48 Obligation: 305.45/291.48 Analyzing the following TRS for decreasing loops: 305.45/291.48 305.45/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.45/291.48 305.45/291.48 305.45/291.48 The TRS R consists of the following rules: 305.45/291.48 305.45/291.48 cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) 305.45/291.48 cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z) 305.45/291.48 cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) 305.45/291.48 cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z) 305.45/291.48 cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z) 305.45/291.48 gr(0, x) -> false 305.45/291.48 gr(s(x), 0) -> true 305.45/291.48 gr(s(x), s(y)) -> gr(x, y) 305.45/291.48 add(0, x) -> x 305.45/291.48 add(s(x), y) -> s(add(x, y)) 305.45/291.48 p(0) -> 0 305.45/291.48 p(s(x)) -> x 305.45/291.48 305.45/291.48 S is empty. 305.45/291.48 Rewrite Strategy: FULL 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (3) DecreasingLoopProof (LOWER BOUND(ID)) 305.45/291.48 The following loop(s) give(s) rise to the lower bound Omega(n^1): 305.45/291.48 305.45/291.48 The rewrite sequence 305.45/291.48 305.45/291.48 add(s(x), y) ->^+ s(add(x, y)) 305.45/291.48 305.45/291.48 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 305.45/291.48 305.45/291.48 The pumping substitution is [x / s(x)]. 305.45/291.48 305.45/291.48 The result substitution is [ ]. 305.45/291.48 305.45/291.48 305.45/291.48 305.45/291.48 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (4) 305.45/291.48 Complex Obligation (BEST) 305.45/291.48 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (5) 305.45/291.48 Obligation: 305.45/291.48 Proved the lower bound n^1 for the following obligation: 305.45/291.48 305.45/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.45/291.48 305.45/291.48 305.45/291.48 The TRS R consists of the following rules: 305.45/291.48 305.45/291.48 cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) 305.45/291.48 cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z) 305.45/291.48 cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) 305.45/291.48 cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z) 305.45/291.48 cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z) 305.45/291.48 gr(0, x) -> false 305.45/291.48 gr(s(x), 0) -> true 305.45/291.48 gr(s(x), s(y)) -> gr(x, y) 305.45/291.48 add(0, x) -> x 305.45/291.48 add(s(x), y) -> s(add(x, y)) 305.45/291.48 p(0) -> 0 305.45/291.48 p(s(x)) -> x 305.45/291.48 305.45/291.48 S is empty. 305.45/291.48 Rewrite Strategy: FULL 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (6) LowerBoundPropagationProof (FINISHED) 305.45/291.48 Propagated lower bound. 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (7) 305.45/291.48 BOUNDS(n^1, INF) 305.45/291.48 305.45/291.48 ---------------------------------------- 305.45/291.48 305.45/291.48 (8) 305.45/291.48 Obligation: 305.45/291.48 Analyzing the following TRS for decreasing loops: 305.45/291.48 305.45/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.45/291.48 305.45/291.48 305.45/291.48 The TRS R consists of the following rules: 305.45/291.48 305.45/291.48 cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) 305.45/291.48 cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z) 305.45/291.48 cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) 305.45/291.48 cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z) 305.45/291.48 cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z) 305.45/291.48 gr(0, x) -> false 305.45/291.48 gr(s(x), 0) -> true 305.45/291.48 gr(s(x), s(y)) -> gr(x, y) 305.45/291.48 add(0, x) -> x 305.45/291.48 add(s(x), y) -> s(add(x, y)) 305.45/291.48 p(0) -> 0 305.45/291.48 p(s(x)) -> x 305.45/291.48 305.45/291.48 S is empty. 305.45/291.48 Rewrite Strategy: FULL 305.56/291.51 EOF