WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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


The TRS R consists of the following rules:

   a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(g(X)) -> g(mark(X))
   a__f(X1, X2) -> f(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(g(X)) -> g(mark(X))
   a__f(X1, X2) -> f(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST
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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

mark(f(X1, X2)) ->^+ a__f(mark(X1), X2)

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

The pumping substitution is [X1 / f(X1, X2)].

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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(g(X)) -> g(mark(X))
   a__f(X1, X2) -> f(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(g(X)) -> g(mark(X))
   a__f(X1, X2) -> f(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST