1115.71/291.53 WORST_CASE(Omega(n^1), ?) 1132.74/295.82 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1132.74/295.82 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1132.74/295.82 1132.74/295.82 1132.74/295.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1132.74/295.82 1132.74/295.82 (0) CpxTRS 1132.74/295.82 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1132.74/295.82 (2) TRS for Loop Detection 1132.74/295.82 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1132.74/295.82 (4) BEST 1132.74/295.82 (5) proven lower bound 1132.74/295.82 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1132.74/295.82 (7) BOUNDS(n^1, INF) 1132.74/295.82 (8) TRS for Loop Detection 1132.74/295.82 1132.74/295.82 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (0) 1132.74/295.82 Obligation: 1132.74/295.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1132.74/295.82 1132.74/295.82 1132.74/295.82 The TRS R consists of the following rules: 1132.74/295.82 1132.74/295.82 minus(minus(x, y), z) -> minus(x, plus(y, z)) 1132.74/295.82 minus(0, y) -> 0 1132.74/295.82 minus(x, 0) -> x 1132.74/295.82 minus(s(x), s(y)) -> minus(x, y) 1132.74/295.82 plus(0, y) -> y 1132.74/295.82 plus(s(x), y) -> plus(x, s(y)) 1132.74/295.82 plus(s(x), y) -> s(plus(y, x)) 1132.74/295.82 zero(s(x)) -> false 1132.74/295.82 zero(0) -> true 1132.74/295.82 p(s(x)) -> x 1132.74/295.82 p(0) -> 0 1132.74/295.82 div(x, y) -> quot(x, y, 0) 1132.74/295.82 quot(s(x), s(y), z) -> quot(minus(p(ack(0, x)), y), s(y), s(z)) 1132.74/295.82 quot(0, s(y), z) -> z 1132.74/295.82 ack(0, x) -> s(x) 1132.74/295.82 ack(0, x) -> plus(x, s(0)) 1132.74/295.82 ack(s(x), 0) -> ack(x, s(0)) 1132.74/295.82 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1132.74/295.82 1132.74/295.82 S is empty. 1132.74/295.82 Rewrite Strategy: FULL 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1132.74/295.82 Transformed a relative TRS into a decreasing-loop problem. 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (2) 1132.74/295.82 Obligation: 1132.74/295.82 Analyzing the following TRS for decreasing loops: 1132.74/295.82 1132.74/295.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1132.74/295.82 1132.74/295.82 1132.74/295.82 The TRS R consists of the following rules: 1132.74/295.82 1132.74/295.82 minus(minus(x, y), z) -> minus(x, plus(y, z)) 1132.74/295.82 minus(0, y) -> 0 1132.74/295.82 minus(x, 0) -> x 1132.74/295.82 minus(s(x), s(y)) -> minus(x, y) 1132.74/295.82 plus(0, y) -> y 1132.74/295.82 plus(s(x), y) -> plus(x, s(y)) 1132.74/295.82 plus(s(x), y) -> s(plus(y, x)) 1132.74/295.82 zero(s(x)) -> false 1132.74/295.82 zero(0) -> true 1132.74/295.82 p(s(x)) -> x 1132.74/295.82 p(0) -> 0 1132.74/295.82 div(x, y) -> quot(x, y, 0) 1132.74/295.82 quot(s(x), s(y), z) -> quot(minus(p(ack(0, x)), y), s(y), s(z)) 1132.74/295.82 quot(0, s(y), z) -> z 1132.74/295.82 ack(0, x) -> s(x) 1132.74/295.82 ack(0, x) -> plus(x, s(0)) 1132.74/295.82 ack(s(x), 0) -> ack(x, s(0)) 1132.74/295.82 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1132.74/295.82 1132.74/295.82 S is empty. 1132.74/295.82 Rewrite Strategy: FULL 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1132.74/295.82 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1132.74/295.82 1132.74/295.82 The rewrite sequence 1132.74/295.82 1132.74/295.82 plus(s(x), y) ->^+ plus(x, s(y)) 1132.74/295.82 1132.74/295.82 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1132.74/295.82 1132.74/295.82 The pumping substitution is [x / s(x)]. 1132.74/295.82 1132.74/295.82 The result substitution is [y / s(y)]. 1132.74/295.82 1132.74/295.82 1132.74/295.82 1132.74/295.82 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (4) 1132.74/295.82 Complex Obligation (BEST) 1132.74/295.82 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (5) 1132.74/295.82 Obligation: 1132.74/295.82 Proved the lower bound n^1 for the following obligation: 1132.74/295.82 1132.74/295.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1132.74/295.82 1132.74/295.82 1132.74/295.82 The TRS R consists of the following rules: 1132.74/295.82 1132.74/295.82 minus(minus(x, y), z) -> minus(x, plus(y, z)) 1132.74/295.82 minus(0, y) -> 0 1132.74/295.82 minus(x, 0) -> x 1132.74/295.82 minus(s(x), s(y)) -> minus(x, y) 1132.74/295.82 plus(0, y) -> y 1132.74/295.82 plus(s(x), y) -> plus(x, s(y)) 1132.74/295.82 plus(s(x), y) -> s(plus(y, x)) 1132.74/295.82 zero(s(x)) -> false 1132.74/295.82 zero(0) -> true 1132.74/295.82 p(s(x)) -> x 1132.74/295.82 p(0) -> 0 1132.74/295.82 div(x, y) -> quot(x, y, 0) 1132.74/295.82 quot(s(x), s(y), z) -> quot(minus(p(ack(0, x)), y), s(y), s(z)) 1132.74/295.82 quot(0, s(y), z) -> z 1132.74/295.82 ack(0, x) -> s(x) 1132.74/295.82 ack(0, x) -> plus(x, s(0)) 1132.74/295.82 ack(s(x), 0) -> ack(x, s(0)) 1132.74/295.82 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1132.74/295.82 1132.74/295.82 S is empty. 1132.74/295.82 Rewrite Strategy: FULL 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (6) LowerBoundPropagationProof (FINISHED) 1132.74/295.82 Propagated lower bound. 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (7) 1132.74/295.82 BOUNDS(n^1, INF) 1132.74/295.82 1132.74/295.82 ---------------------------------------- 1132.74/295.82 1132.74/295.82 (8) 1132.74/295.82 Obligation: 1132.74/295.82 Analyzing the following TRS for decreasing loops: 1132.74/295.82 1132.74/295.82 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1132.74/295.82 1132.74/295.82 1132.74/295.82 The TRS R consists of the following rules: 1132.74/295.82 1132.74/295.82 minus(minus(x, y), z) -> minus(x, plus(y, z)) 1132.74/295.82 minus(0, y) -> 0 1132.74/295.82 minus(x, 0) -> x 1132.74/295.82 minus(s(x), s(y)) -> minus(x, y) 1132.74/295.82 plus(0, y) -> y 1132.74/295.82 plus(s(x), y) -> plus(x, s(y)) 1132.74/295.82 plus(s(x), y) -> s(plus(y, x)) 1132.74/295.82 zero(s(x)) -> false 1132.74/295.82 zero(0) -> true 1132.74/295.82 p(s(x)) -> x 1132.74/295.82 p(0) -> 0 1132.74/295.82 div(x, y) -> quot(x, y, 0) 1132.74/295.82 quot(s(x), s(y), z) -> quot(minus(p(ack(0, x)), y), s(y), s(z)) 1132.74/295.82 quot(0, s(y), z) -> z 1132.74/295.82 ack(0, x) -> s(x) 1132.74/295.82 ack(0, x) -> plus(x, s(0)) 1132.74/295.82 ack(s(x), 0) -> ack(x, s(0)) 1132.74/295.82 ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 1132.74/295.82 1132.74/295.82 S is empty. 1132.74/295.82 Rewrite Strategy: FULL 1133.06/295.89 EOF