311.32/291.57 WORST_CASE(Omega(n^1), ?) 311.32/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 311.32/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 311.32/291.58 311.32/291.58 311.32/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.32/291.58 311.32/291.58 (0) CpxTRS 311.32/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 311.32/291.58 (2) TRS for Loop Detection 311.32/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 311.32/291.58 (4) BEST 311.32/291.58 (5) proven lower bound 311.32/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 311.32/291.58 (7) BOUNDS(n^1, INF) 311.32/291.58 (8) TRS for Loop Detection 311.32/291.58 311.32/291.58 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (0) 311.32/291.58 Obligation: 311.32/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.32/291.58 311.32/291.58 311.32/291.58 The TRS R consists of the following rules: 311.32/291.58 311.32/291.58 inc(s(x)) -> s(inc(x)) 311.32/291.58 inc(0) -> s(0) 311.32/291.58 plus(x, y) -> ifPlus(eq(x, 0), minus(x, s(0)), x, inc(x)) 311.32/291.58 ifPlus(false, x, y, z) -> plus(x, z) 311.32/291.58 ifPlus(true, x, y, z) -> y 311.32/291.58 minus(s(x), s(y)) -> minus(x, y) 311.32/291.58 minus(0, x) -> 0 311.32/291.58 minus(x, 0) -> x 311.32/291.58 minus(x, x) -> 0 311.32/291.58 eq(s(x), s(y)) -> eq(x, y) 311.32/291.58 eq(0, s(x)) -> false 311.32/291.58 eq(s(x), 0) -> false 311.32/291.58 eq(0, 0) -> true 311.32/291.58 eq(x, x) -> true 311.32/291.58 times(x, y) -> timesIter(x, y, 0) 311.32/291.58 timesIter(x, y, z) -> ifTimes(eq(x, 0), minus(x, s(0)), y, z, plus(y, z)) 311.32/291.58 ifTimes(true, x, y, z, u) -> z 311.32/291.58 ifTimes(false, x, y, z, u) -> timesIter(x, y, u) 311.32/291.58 f -> g 311.32/291.58 f -> h 311.32/291.58 311.32/291.58 S is empty. 311.32/291.58 Rewrite Strategy: FULL 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 311.32/291.58 Transformed a relative TRS into a decreasing-loop problem. 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (2) 311.32/291.58 Obligation: 311.32/291.58 Analyzing the following TRS for decreasing loops: 311.32/291.58 311.32/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.32/291.58 311.32/291.58 311.32/291.58 The TRS R consists of the following rules: 311.32/291.58 311.32/291.58 inc(s(x)) -> s(inc(x)) 311.32/291.58 inc(0) -> s(0) 311.32/291.58 plus(x, y) -> ifPlus(eq(x, 0), minus(x, s(0)), x, inc(x)) 311.32/291.58 ifPlus(false, x, y, z) -> plus(x, z) 311.32/291.58 ifPlus(true, x, y, z) -> y 311.32/291.58 minus(s(x), s(y)) -> minus(x, y) 311.32/291.58 minus(0, x) -> 0 311.32/291.58 minus(x, 0) -> x 311.32/291.58 minus(x, x) -> 0 311.32/291.58 eq(s(x), s(y)) -> eq(x, y) 311.32/291.58 eq(0, s(x)) -> false 311.32/291.58 eq(s(x), 0) -> false 311.32/291.58 eq(0, 0) -> true 311.32/291.58 eq(x, x) -> true 311.32/291.58 times(x, y) -> timesIter(x, y, 0) 311.32/291.58 timesIter(x, y, z) -> ifTimes(eq(x, 0), minus(x, s(0)), y, z, plus(y, z)) 311.32/291.58 ifTimes(true, x, y, z, u) -> z 311.32/291.58 ifTimes(false, x, y, z, u) -> timesIter(x, y, u) 311.32/291.58 f -> g 311.32/291.58 f -> h 311.32/291.58 311.32/291.58 S is empty. 311.32/291.58 Rewrite Strategy: FULL 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 311.32/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 311.32/291.58 311.32/291.58 The rewrite sequence 311.32/291.58 311.32/291.58 minus(s(x), s(y)) ->^+ minus(x, y) 311.32/291.58 311.32/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 311.32/291.58 311.32/291.58 The pumping substitution is [x / s(x), y / s(y)]. 311.32/291.58 311.32/291.58 The result substitution is [ ]. 311.32/291.58 311.32/291.58 311.32/291.58 311.32/291.58 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (4) 311.32/291.58 Complex Obligation (BEST) 311.32/291.58 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (5) 311.32/291.58 Obligation: 311.32/291.58 Proved the lower bound n^1 for the following obligation: 311.32/291.58 311.32/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.32/291.58 311.32/291.58 311.32/291.58 The TRS R consists of the following rules: 311.32/291.58 311.32/291.58 inc(s(x)) -> s(inc(x)) 311.32/291.58 inc(0) -> s(0) 311.32/291.58 plus(x, y) -> ifPlus(eq(x, 0), minus(x, s(0)), x, inc(x)) 311.32/291.58 ifPlus(false, x, y, z) -> plus(x, z) 311.32/291.58 ifPlus(true, x, y, z) -> y 311.32/291.58 minus(s(x), s(y)) -> minus(x, y) 311.32/291.58 minus(0, x) -> 0 311.32/291.58 minus(x, 0) -> x 311.32/291.58 minus(x, x) -> 0 311.32/291.58 eq(s(x), s(y)) -> eq(x, y) 311.32/291.58 eq(0, s(x)) -> false 311.32/291.58 eq(s(x), 0) -> false 311.32/291.58 eq(0, 0) -> true 311.32/291.58 eq(x, x) -> true 311.32/291.58 times(x, y) -> timesIter(x, y, 0) 311.32/291.58 timesIter(x, y, z) -> ifTimes(eq(x, 0), minus(x, s(0)), y, z, plus(y, z)) 311.32/291.58 ifTimes(true, x, y, z, u) -> z 311.32/291.58 ifTimes(false, x, y, z, u) -> timesIter(x, y, u) 311.32/291.58 f -> g 311.32/291.58 f -> h 311.32/291.58 311.32/291.58 S is empty. 311.32/291.58 Rewrite Strategy: FULL 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (6) LowerBoundPropagationProof (FINISHED) 311.32/291.58 Propagated lower bound. 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (7) 311.32/291.58 BOUNDS(n^1, INF) 311.32/291.58 311.32/291.58 ---------------------------------------- 311.32/291.58 311.32/291.58 (8) 311.32/291.58 Obligation: 311.32/291.58 Analyzing the following TRS for decreasing loops: 311.32/291.58 311.32/291.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.32/291.58 311.32/291.58 311.32/291.58 The TRS R consists of the following rules: 311.32/291.58 311.32/291.58 inc(s(x)) -> s(inc(x)) 311.32/291.58 inc(0) -> s(0) 311.32/291.58 plus(x, y) -> ifPlus(eq(x, 0), minus(x, s(0)), x, inc(x)) 311.32/291.58 ifPlus(false, x, y, z) -> plus(x, z) 311.32/291.58 ifPlus(true, x, y, z) -> y 311.32/291.58 minus(s(x), s(y)) -> minus(x, y) 311.32/291.58 minus(0, x) -> 0 311.32/291.58 minus(x, 0) -> x 311.32/291.58 minus(x, x) -> 0 311.32/291.58 eq(s(x), s(y)) -> eq(x, y) 311.32/291.58 eq(0, s(x)) -> false 311.32/291.58 eq(s(x), 0) -> false 311.32/291.58 eq(0, 0) -> true 311.32/291.58 eq(x, x) -> true 311.32/291.58 times(x, y) -> timesIter(x, y, 0) 311.32/291.58 timesIter(x, y, z) -> ifTimes(eq(x, 0), minus(x, s(0)), y, z, plus(y, z)) 311.32/291.58 ifTimes(true, x, y, z, u) -> z 311.32/291.58 ifTimes(false, x, y, z, u) -> timesIter(x, y, u) 311.32/291.58 f -> g 311.32/291.58 f -> h 311.32/291.58 311.32/291.58 S is empty. 311.32/291.58 Rewrite Strategy: FULL 311.32/291.60 EOF