/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
   ifPlus(true, x, y) -> p(y)
   ifPlus(false, x, y) -> plus(p(x), y)
   times(x, y) -> timesIter(0, x, y, 0)
   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
   ifTimes(true, i, x, y, z) -> z
   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))
   inc(x) -> s(x)
   p(0) -> 0
   p(s(x)) -> x
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   f0(0, y, x) -> f1(x, y, x)
   f1(x, y, z) -> f2(x, y, z)
   f2(x, 1, z) -> f0(x, z, z)
   f0(x, y, z) -> d
   f1(x, y, z) -> c

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(2)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
   ifPlus(true, x, y) -> p(y)
   ifPlus(false, x, y) -> plus(p(x), y)
   times(x, y) -> timesIter(0, x, y, 0)
   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
   ifTimes(true, i, x, y, z) -> z
   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))
   inc(x) -> s(x)
   p(0) -> 0
   p(s(x)) -> x
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   f0(0, y, x) -> f1(x, y, x)
   f1(x, y, z) -> f2(x, y, z)
   f2(x, 1, z) -> f0(x, z, z)
   f0(x, y, z) -> d
   f1(x, y, z) -> c

S is empty.
Rewrite Strategy: FULL
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(3) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

isZero(s(s(x))) ->^+ isZero(s(x))

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x / s(x)].

The result substitution is [ ].




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(4)
Complex Obligation (BEST)

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(5)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
   ifPlus(true, x, y) -> p(y)
   ifPlus(false, x, y) -> plus(p(x), y)
   times(x, y) -> timesIter(0, x, y, 0)
   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
   ifTimes(true, i, x, y, z) -> z
   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))
   inc(x) -> s(x)
   p(0) -> 0
   p(s(x)) -> x
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   f0(0, y, x) -> f1(x, y, x)
   f1(x, y, z) -> f2(x, y, z)
   f2(x, 1, z) -> f0(x, z, z)
   f0(x, y, z) -> d
   f1(x, y, z) -> c

S is empty.
Rewrite Strategy: FULL
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
   ifPlus(true, x, y) -> p(y)
   ifPlus(false, x, y) -> plus(p(x), y)
   times(x, y) -> timesIter(0, x, y, 0)
   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
   ifTimes(true, i, x, y, z) -> z
   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))
   inc(x) -> s(x)
   p(0) -> 0
   p(s(x)) -> x
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   f0(0, y, x) -> f1(x, y, x)
   f1(x, y, z) -> f2(x, y, z)
   f2(x, 1, z) -> f0(x, z, z)
   f0(x, y, z) -> d
   f1(x, y, z) -> c

S is empty.
Rewrite Strategy: FULL