306.36/291.50 WORST_CASE(Omega(n^1), ?) 306.36/291.50 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 306.36/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 306.36/291.50 306.36/291.50 306.36/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.36/291.50 306.36/291.50 (0) CpxTRS 306.36/291.50 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 306.36/291.50 (2) TRS for Loop Detection 306.36/291.50 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 306.36/291.50 (4) BEST 306.36/291.50 (5) proven lower bound 306.36/291.50 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 306.36/291.50 (7) BOUNDS(n^1, INF) 306.36/291.50 (8) TRS for Loop Detection 306.36/291.50 306.36/291.50 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (0) 306.36/291.50 Obligation: 306.36/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.36/291.50 306.36/291.50 306.36/291.50 The TRS R consists of the following rules: 306.36/291.50 306.36/291.50 function(iszero, 0, dummy, dummy2) -> true 306.36/291.50 function(iszero, s(x), dummy, dummy2) -> false 306.36/291.50 function(p, 0, dummy, dummy2) -> 0 306.36/291.50 function(p, s(0), dummy, dummy2) -> 0 306.36/291.50 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) 306.36/291.50 function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) 306.36/291.50 function(if, true, x, y) -> y 306.36/291.50 function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) 306.36/291.50 function(third, x, y, z) -> z 306.36/291.50 306.36/291.50 S is empty. 306.36/291.50 Rewrite Strategy: FULL 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 306.36/291.50 Transformed a relative TRS into a decreasing-loop problem. 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (2) 306.36/291.50 Obligation: 306.36/291.50 Analyzing the following TRS for decreasing loops: 306.36/291.50 306.36/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.36/291.50 306.36/291.50 306.36/291.50 The TRS R consists of the following rules: 306.36/291.50 306.36/291.50 function(iszero, 0, dummy, dummy2) -> true 306.36/291.50 function(iszero, s(x), dummy, dummy2) -> false 306.36/291.50 function(p, 0, dummy, dummy2) -> 0 306.36/291.50 function(p, s(0), dummy, dummy2) -> 0 306.36/291.50 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) 306.36/291.50 function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) 306.36/291.50 function(if, true, x, y) -> y 306.36/291.50 function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) 306.36/291.50 function(third, x, y, z) -> z 306.36/291.50 306.36/291.50 S is empty. 306.36/291.50 Rewrite Strategy: FULL 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (3) DecreasingLoopProof (LOWER BOUND(ID)) 306.36/291.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 306.36/291.50 306.36/291.50 The rewrite sequence 306.36/291.50 306.36/291.50 function(p, s(s(x)), dummy, dummy2) ->^+ s(function(p, s(x), x, x)) 306.36/291.50 306.36/291.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 306.36/291.50 306.36/291.50 The pumping substitution is [x / s(x)]. 306.36/291.50 306.36/291.50 The result substitution is [dummy / x, dummy2 / x]. 306.36/291.50 306.36/291.50 306.36/291.50 306.36/291.50 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (4) 306.36/291.50 Complex Obligation (BEST) 306.36/291.50 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (5) 306.36/291.50 Obligation: 306.36/291.50 Proved the lower bound n^1 for the following obligation: 306.36/291.50 306.36/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.36/291.50 306.36/291.50 306.36/291.50 The TRS R consists of the following rules: 306.36/291.50 306.36/291.50 function(iszero, 0, dummy, dummy2) -> true 306.36/291.50 function(iszero, s(x), dummy, dummy2) -> false 306.36/291.50 function(p, 0, dummy, dummy2) -> 0 306.36/291.50 function(p, s(0), dummy, dummy2) -> 0 306.36/291.50 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) 306.36/291.50 function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) 306.36/291.50 function(if, true, x, y) -> y 306.36/291.50 function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) 306.36/291.50 function(third, x, y, z) -> z 306.36/291.50 306.36/291.50 S is empty. 306.36/291.50 Rewrite Strategy: FULL 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (6) LowerBoundPropagationProof (FINISHED) 306.36/291.50 Propagated lower bound. 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (7) 306.36/291.50 BOUNDS(n^1, INF) 306.36/291.50 306.36/291.50 ---------------------------------------- 306.36/291.50 306.36/291.50 (8) 306.36/291.50 Obligation: 306.36/291.50 Analyzing the following TRS for decreasing loops: 306.36/291.50 306.36/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.36/291.50 306.36/291.50 306.36/291.50 The TRS R consists of the following rules: 306.36/291.50 306.36/291.50 function(iszero, 0, dummy, dummy2) -> true 306.36/291.50 function(iszero, s(x), dummy, dummy2) -> false 306.36/291.50 function(p, 0, dummy, dummy2) -> 0 306.36/291.50 function(p, s(0), dummy, dummy2) -> 0 306.36/291.50 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) 306.36/291.50 function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) 306.36/291.50 function(if, true, x, y) -> y 306.36/291.50 function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) 306.36/291.50 function(third, x, y, z) -> z 306.36/291.50 306.36/291.50 S is empty. 306.36/291.50 Rewrite Strategy: FULL 306.36/291.55 EOF