/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   top1(free(x), y) -> top2(check(new(x)), y)
   top1(free(x), y) -> top2(new(x), check(y))
   top1(free(x), y) -> top2(check(x), new(y))
   top1(free(x), y) -> top2(x, check(new(y)))
   top2(x, free(y)) -> top1(check(new(x)), y)
   top2(x, free(y)) -> top1(new(x), check(y))
   top2(x, free(y)) -> top1(check(x), new(y))
   top2(x, free(y)) -> top1(x, check(new(y)))
   new(free(x)) -> free(new(x))
   old(free(x)) -> free(old(x))
   new(serve) -> free(serve)
   old(serve) -> free(serve)
   check(free(x)) -> free(check(x))
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> old(x)

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:

   top1(free(x), y) -> top2(check(new(x)), y)
   top1(free(x), y) -> top2(new(x), check(y))
   top1(free(x), y) -> top2(check(x), new(y))
   top1(free(x), y) -> top2(x, check(new(y)))
   top2(x, free(y)) -> top1(check(new(x)), y)
   top2(x, free(y)) -> top1(new(x), check(y))
   top2(x, free(y)) -> top1(check(x), new(y))
   top2(x, free(y)) -> top1(x, check(new(y)))
   new(free(x)) -> free(new(x))
   old(free(x)) -> free(old(x))
   new(serve) -> free(serve)
   old(serve) -> free(serve)
   check(free(x)) -> free(check(x))
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> old(x)

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

check(free(x)) ->^+ free(check(x))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / free(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:

   top1(free(x), y) -> top2(check(new(x)), y)
   top1(free(x), y) -> top2(new(x), check(y))
   top1(free(x), y) -> top2(check(x), new(y))
   top1(free(x), y) -> top2(x, check(new(y)))
   top2(x, free(y)) -> top1(check(new(x)), y)
   top2(x, free(y)) -> top1(new(x), check(y))
   top2(x, free(y)) -> top1(check(x), new(y))
   top2(x, free(y)) -> top1(x, check(new(y)))
   new(free(x)) -> free(new(x))
   old(free(x)) -> free(old(x))
   new(serve) -> free(serve)
   old(serve) -> free(serve)
   check(free(x)) -> free(check(x))
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> old(x)

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:

   top1(free(x), y) -> top2(check(new(x)), y)
   top1(free(x), y) -> top2(new(x), check(y))
   top1(free(x), y) -> top2(check(x), new(y))
   top1(free(x), y) -> top2(x, check(new(y)))
   top2(x, free(y)) -> top1(check(new(x)), y)
   top2(x, free(y)) -> top1(new(x), check(y))
   top2(x, free(y)) -> top1(check(x), new(y))
   top2(x, free(y)) -> top1(x, check(new(y)))
   new(free(x)) -> free(new(x))
   old(free(x)) -> free(old(x))
   new(serve) -> free(serve)
   old(serve) -> free(serve)
   check(free(x)) -> free(check(x))
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> old(x)

S is empty.
Rewrite Strategy: FULL