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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^1).

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


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: FULL
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(3) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
The certificate found is represented by the following graph.

"[1, 2, 5, 6, 7, 8]
{(1,2,[a__f_1|0, mark_1|0, f_1|1, a__f_1|1, a|1, f_1|2]), (1,5,[a__f_1|1, f_1|2]), (1,8,[g_1|1]), (2,2,[f_1|0, a|0, g_1|0]), (5,6,[g_1|1]), (6,7,[f_1|1]), (7,2,[a|1]), (8,2,[mark_1|1, a__f_1|1, a|1, f_1|2]), (8,8,[g_1|1]), (8,5,[a__f_1|1, f_1|2])}"
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(4)
BOUNDS(1, n^1)

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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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

The rewrite sequence

mark(g(X)) ->^+ g(mark(X))

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

The pumping substitution is [X / g(X)].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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(9)
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, n^1).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: FULL
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: FULL