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


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [SOUND, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
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)

The set Q consists of the following terms:

   mark(f(x0))
   mark(a)
   mark(g(x0))
   a__f(x0)


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(1) QTRS Reverse (SOUND)
We applied the QTRS Reverse Processor [REVERSE].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(3) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Node 11 is start node and node 12 is final node.

Those nodes are connected through the following edges:

* 11 to 13 labelled a'_1(0)* 11 to 12 labelled a__f_1(0), a'_1(0), f_1(0), f_1(1), a'_1(1), a__f_1(1), f_1(2)* 11 to 16 labelled mark_1(0)* 11 to 17 labelled a'_1(1)* 12 to 12 labelled #_1(0)* 13 to 14 labelled f_1(0)* 14 to 15 labelled g_1(0)* 15 to 12 labelled a__f_1(0), f_1(1), a__f_1(1), f_1(2)* 16 to 12 labelled g_1(0)* 16 to 20 labelled mark_1(1)* 17 to 18 labelled f_1(1)* 18 to 19 labelled g_1(1)* 19 to 12 labelled a__f_1(1), f_1(2)* 20 to 12 labelled g_1(1)* 20 to 20 labelled mark_1(1)


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(4)
YES