1106.00/291.47 WORST_CASE(Omega(n^1), ?) 1106.00/291.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1106.00/291.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1106.00/291.48 1106.00/291.48 1106.00/291.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1106.00/291.48 1106.00/291.48 (0) CpxTRS 1106.00/291.48 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1106.00/291.48 (2) TRS for Loop Detection 1106.00/291.48 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1106.00/291.48 (4) BEST 1106.00/291.48 (5) proven lower bound 1106.00/291.48 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1106.00/291.48 (7) BOUNDS(n^1, INF) 1106.00/291.48 (8) TRS for Loop Detection 1106.00/291.48 1106.00/291.48 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (0) 1106.00/291.48 Obligation: 1106.00/291.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1106.00/291.48 1106.00/291.48 1106.00/291.48 The TRS R consists of the following rules: 1106.00/291.48 1106.00/291.48 prod(xs) -> prodIter(xs, s(0)) 1106.00/291.48 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 1106.00/291.48 ifProd(true, xs, x) -> x 1106.00/291.48 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 1106.00/291.48 plus(0, y) -> y 1106.00/291.48 plus(s(x), y) -> s(plus(x, y)) 1106.00/291.48 times(x, y) -> timesIter(x, y, 0, 0) 1106.00/291.48 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 1106.00/291.48 ifTimes(true, x, y, z, u) -> z 1106.00/291.48 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 1106.00/291.48 isempty(nil) -> true 1106.00/291.48 isempty(cons(x, xs)) -> false 1106.00/291.48 head(nil) -> error 1106.00/291.48 head(cons(x, xs)) -> x 1106.00/291.48 tail(nil) -> nil 1106.00/291.48 tail(cons(x, xs)) -> xs 1106.00/291.48 ge(x, 0) -> true 1106.00/291.48 ge(0, s(y)) -> false 1106.00/291.48 ge(s(x), s(y)) -> ge(x, y) 1106.00/291.48 a -> b 1106.00/291.48 a -> c 1106.00/291.48 1106.00/291.48 S is empty. 1106.00/291.48 Rewrite Strategy: INNERMOST 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1106.00/291.48 Transformed a relative TRS into a decreasing-loop problem. 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (2) 1106.00/291.48 Obligation: 1106.00/291.48 Analyzing the following TRS for decreasing loops: 1106.00/291.48 1106.00/291.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1106.00/291.48 1106.00/291.48 1106.00/291.48 The TRS R consists of the following rules: 1106.00/291.48 1106.00/291.48 prod(xs) -> prodIter(xs, s(0)) 1106.00/291.48 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 1106.00/291.48 ifProd(true, xs, x) -> x 1106.00/291.48 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 1106.00/291.48 plus(0, y) -> y 1106.00/291.48 plus(s(x), y) -> s(plus(x, y)) 1106.00/291.48 times(x, y) -> timesIter(x, y, 0, 0) 1106.00/291.48 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 1106.00/291.48 ifTimes(true, x, y, z, u) -> z 1106.00/291.48 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 1106.00/291.48 isempty(nil) -> true 1106.00/291.48 isempty(cons(x, xs)) -> false 1106.00/291.48 head(nil) -> error 1106.00/291.48 head(cons(x, xs)) -> x 1106.00/291.48 tail(nil) -> nil 1106.00/291.48 tail(cons(x, xs)) -> xs 1106.00/291.48 ge(x, 0) -> true 1106.00/291.48 ge(0, s(y)) -> false 1106.00/291.48 ge(s(x), s(y)) -> ge(x, y) 1106.00/291.48 a -> b 1106.00/291.48 a -> c 1106.00/291.48 1106.00/291.48 S is empty. 1106.00/291.48 Rewrite Strategy: INNERMOST 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1106.00/291.48 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1106.00/291.48 1106.00/291.48 The rewrite sequence 1106.00/291.48 1106.00/291.48 plus(s(x), y) ->^+ s(plus(x, y)) 1106.00/291.48 1106.00/291.48 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1106.00/291.48 1106.00/291.48 The pumping substitution is [x / s(x)]. 1106.00/291.48 1106.00/291.48 The result substitution is [ ]. 1106.00/291.48 1106.00/291.48 1106.00/291.48 1106.00/291.48 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (4) 1106.00/291.48 Complex Obligation (BEST) 1106.00/291.48 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (5) 1106.00/291.48 Obligation: 1106.00/291.48 Proved the lower bound n^1 for the following obligation: 1106.00/291.48 1106.00/291.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1106.00/291.48 1106.00/291.48 1106.00/291.48 The TRS R consists of the following rules: 1106.00/291.48 1106.00/291.48 prod(xs) -> prodIter(xs, s(0)) 1106.00/291.48 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 1106.00/291.48 ifProd(true, xs, x) -> x 1106.00/291.48 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 1106.00/291.48 plus(0, y) -> y 1106.00/291.48 plus(s(x), y) -> s(plus(x, y)) 1106.00/291.48 times(x, y) -> timesIter(x, y, 0, 0) 1106.00/291.48 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 1106.00/291.48 ifTimes(true, x, y, z, u) -> z 1106.00/291.48 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 1106.00/291.48 isempty(nil) -> true 1106.00/291.48 isempty(cons(x, xs)) -> false 1106.00/291.48 head(nil) -> error 1106.00/291.48 head(cons(x, xs)) -> x 1106.00/291.48 tail(nil) -> nil 1106.00/291.48 tail(cons(x, xs)) -> xs 1106.00/291.48 ge(x, 0) -> true 1106.00/291.48 ge(0, s(y)) -> false 1106.00/291.48 ge(s(x), s(y)) -> ge(x, y) 1106.00/291.48 a -> b 1106.00/291.48 a -> c 1106.00/291.48 1106.00/291.48 S is empty. 1106.00/291.48 Rewrite Strategy: INNERMOST 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (6) LowerBoundPropagationProof (FINISHED) 1106.00/291.48 Propagated lower bound. 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (7) 1106.00/291.48 BOUNDS(n^1, INF) 1106.00/291.48 1106.00/291.48 ---------------------------------------- 1106.00/291.48 1106.00/291.48 (8) 1106.00/291.48 Obligation: 1106.00/291.48 Analyzing the following TRS for decreasing loops: 1106.00/291.48 1106.00/291.48 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1106.00/291.48 1106.00/291.48 1106.00/291.48 The TRS R consists of the following rules: 1106.00/291.48 1106.00/291.48 prod(xs) -> prodIter(xs, s(0)) 1106.00/291.48 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 1106.00/291.48 ifProd(true, xs, x) -> x 1106.00/291.48 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 1106.00/291.48 plus(0, y) -> y 1106.00/291.48 plus(s(x), y) -> s(plus(x, y)) 1106.00/291.48 times(x, y) -> timesIter(x, y, 0, 0) 1106.00/291.48 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 1106.00/291.48 ifTimes(true, x, y, z, u) -> z 1106.00/291.48 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 1106.00/291.48 isempty(nil) -> true 1106.00/291.48 isempty(cons(x, xs)) -> false 1106.00/291.48 head(nil) -> error 1106.00/291.48 head(cons(x, xs)) -> x 1106.00/291.48 tail(nil) -> nil 1106.00/291.48 tail(cons(x, xs)) -> xs 1106.00/291.48 ge(x, 0) -> true 1106.00/291.48 ge(0, s(y)) -> false 1106.00/291.48 ge(s(x), s(y)) -> ge(x, y) 1106.00/291.48 a -> b 1106.00/291.48 a -> c 1106.00/291.48 1106.00/291.48 S is empty. 1106.00/291.48 Rewrite Strategy: INNERMOST 1106.00/291.54 EOF