305.69/291.53 WORST_CASE(Omega(n^1), ?) 305.84/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 305.84/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.84/291.58 305.84/291.58 305.84/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.84/291.58 305.84/291.58 (0) CpxTRS 305.84/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 305.84/291.58 (2) TRS for Loop Detection 305.84/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 305.84/291.58 (4) BEST 305.84/291.58 (5) proven lower bound 305.84/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 305.84/291.58 (7) BOUNDS(n^1, INF) 305.84/291.58 (8) TRS for Loop Detection 305.84/291.58 305.84/291.58 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (0) 305.84/291.58 Obligation: 305.84/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.84/291.58 305.84/291.58 305.84/291.58 The TRS R consists of the following rules: 305.84/291.58 305.84/291.58 ge(x, 0) -> true 305.84/291.58 ge(0, s(y)) -> false 305.84/291.58 ge(s(x), s(y)) -> ge(x, y) 305.84/291.58 minus(x, 0) -> x 305.84/291.58 minus(0, y) -> 0 305.84/291.58 minus(s(x), s(y)) -> minus(x, y) 305.84/291.58 id_inc(x) -> x 305.84/291.58 id_inc(x) -> s(x) 305.84/291.58 quot(x, y) -> div(x, y, 0) 305.84/291.58 div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z) 305.84/291.58 if(false, b, x, y, z) -> div_by_zero 305.84/291.58 if(true, false, x, y, z) -> z 305.84/291.58 if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z)) 305.84/291.58 305.84/291.58 S is empty. 305.84/291.58 Rewrite Strategy: FULL 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 305.84/291.58 Transformed a relative TRS into a decreasing-loop problem. 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (2) 305.84/291.58 Obligation: 305.84/291.58 Analyzing the following TRS for decreasing loops: 305.84/291.58 305.84/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.84/291.58 305.84/291.58 305.84/291.58 The TRS R consists of the following rules: 305.84/291.58 305.84/291.58 ge(x, 0) -> true 305.84/291.58 ge(0, s(y)) -> false 305.84/291.58 ge(s(x), s(y)) -> ge(x, y) 305.84/291.58 minus(x, 0) -> x 305.84/291.58 minus(0, y) -> 0 305.84/291.58 minus(s(x), s(y)) -> minus(x, y) 305.84/291.58 id_inc(x) -> x 305.84/291.58 id_inc(x) -> s(x) 305.84/291.58 quot(x, y) -> div(x, y, 0) 305.84/291.58 div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z) 305.84/291.58 if(false, b, x, y, z) -> div_by_zero 305.84/291.58 if(true, false, x, y, z) -> z 305.84/291.58 if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z)) 305.84/291.58 305.84/291.58 S is empty. 305.84/291.58 Rewrite Strategy: FULL 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 305.84/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 305.84/291.58 305.84/291.58 The rewrite sequence 305.84/291.58 305.84/291.58 minus(s(x), s(y)) ->^+ minus(x, y) 305.84/291.58 305.84/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 305.84/291.58 305.84/291.58 The pumping substitution is [x / s(x), y / s(y)]. 305.84/291.58 305.84/291.58 The result substitution is [ ]. 305.84/291.58 305.84/291.58 305.84/291.58 305.84/291.58 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (4) 305.84/291.58 Complex Obligation (BEST) 305.84/291.58 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (5) 305.84/291.58 Obligation: 305.84/291.58 Proved the lower bound n^1 for the following obligation: 305.84/291.58 305.84/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.84/291.58 305.84/291.58 305.84/291.58 The TRS R consists of the following rules: 305.84/291.58 305.84/291.58 ge(x, 0) -> true 305.84/291.58 ge(0, s(y)) -> false 305.84/291.58 ge(s(x), s(y)) -> ge(x, y) 305.84/291.58 minus(x, 0) -> x 305.84/291.58 minus(0, y) -> 0 305.84/291.58 minus(s(x), s(y)) -> minus(x, y) 305.84/291.58 id_inc(x) -> x 305.84/291.58 id_inc(x) -> s(x) 305.84/291.58 quot(x, y) -> div(x, y, 0) 305.84/291.58 div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z) 305.84/291.58 if(false, b, x, y, z) -> div_by_zero 305.84/291.58 if(true, false, x, y, z) -> z 305.84/291.58 if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z)) 305.84/291.58 305.84/291.58 S is empty. 305.84/291.58 Rewrite Strategy: FULL 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (6) LowerBoundPropagationProof (FINISHED) 305.84/291.58 Propagated lower bound. 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (7) 305.84/291.58 BOUNDS(n^1, INF) 305.84/291.58 305.84/291.58 ---------------------------------------- 305.84/291.58 305.84/291.58 (8) 305.84/291.58 Obligation: 305.84/291.58 Analyzing the following TRS for decreasing loops: 305.84/291.58 305.84/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.84/291.58 305.84/291.58 305.84/291.58 The TRS R consists of the following rules: 305.84/291.58 305.84/291.58 ge(x, 0) -> true 305.84/291.58 ge(0, s(y)) -> false 305.84/291.58 ge(s(x), s(y)) -> ge(x, y) 305.84/291.58 minus(x, 0) -> x 305.84/291.58 minus(0, y) -> 0 305.84/291.58 minus(s(x), s(y)) -> minus(x, y) 305.84/291.58 id_inc(x) -> x 305.84/291.58 id_inc(x) -> s(x) 305.84/291.58 quot(x, y) -> div(x, y, 0) 305.84/291.58 div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z) 305.84/291.58 if(false, b, x, y, z) -> div_by_zero 305.84/291.58 if(true, false, x, y, z) -> z 305.84/291.58 if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z)) 305.84/291.58 305.84/291.58 S is empty. 305.84/291.58 Rewrite Strategy: FULL 305.84/291.60 EOF