/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), O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 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:

   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(activate(X))
   activate(n__a) -> a
   activate(X) -> 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:

   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(activate(X))
   activate(n__a) -> a
   activate(X) -> 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.

"[17, 18, 19, 20, 21, 22, 23, 24]
{(17,18,[f_1|0, g_1|0, a|0, activate_1|0, n__f_1|1, n__g_1|1, n__a|1, a|1, c_1|1, n__a|2]), (17,19,[f_1|1, n__f_1|2]), (17,20,[g_1|1, n__g_1|2]), (17,21,[c_1|2]), (18,18,[c_1|0, n__f_1|0, n__g_1|0, n__a|0]), (19,18,[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]), (19,19,[f_1|1, n__f_1|2]), (19,20,[g_1|1, n__g_1|2]), (19,21,[c_1|2]), (20,18,[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]), (20,19,[f_1|1, n__f_1|2]), (20,20,[g_1|1, n__g_1|2]), (20,21,[c_1|2]), (21,22,[n__f_1|2]), (22,23,[n__g_1|2]), (23,24,[n__f_1|2]), (24,18,[n__a|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:

   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(activate(X))
   activate(n__a) -> a
   activate(X) -> 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

activate(n__f(X)) ->^+ f(activate(X))

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

The pumping substitution is [X / n__f(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:

   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(activate(X))
   activate(n__a) -> a
   activate(X) -> 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:

   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(activate(X))
   activate(n__a) -> a
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL