YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) Strip Symbols Proof [SOUND, 0 ms]
(4) QTRS
(5) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(b(b(a(a(a(a(x1))))))) -> a(a(a(a(a(b(b(a(a(b(b(x1)))))))))))

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
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(a(a(a(b(b(a(x1))))))) -> b(b(a(a(b(b(a(a(a(a(a(x1)))))))))))

Q is empty.

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(3) Strip Symbols Proof (SOUND)
We were given the following TRS:

   a(a(a(a(b(b(a(x1))))))) -> b(b(a(a(b(b(a(a(a(a(a(x1)))))))))))

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]: 

   a(a(a(a(b(b(x)))))) -> b(b(a(a(b(b(a(a(a(a(x))))))))))

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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(a(a(b(b(x)))))) -> b(b(a(a(b(b(a(a(a(a(x))))))))))

Q is empty.

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(5) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

   a(a(a(a(b(b(x)))))) -> b(b(a(a(b(b(a(a(a(a(x))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508

Node 489 is start node and node 490 is final node.

Those nodes are connected through the following edges:

* 489 to 491 labelled b_1(0)* 490 to 490 labelled #_1(0)* 491 to 492 labelled b_1(0)* 492 to 493 labelled a_1(0)* 493 to 494 labelled a_1(0)* 494 to 495 labelled b_1(0)* 495 to 496 labelled b_1(0)* 496 to 497 labelled a_1(0)* 496 to 500 labelled b_1(1)* 497 to 498 labelled a_1(0)* 497 to 500 labelled b_1(1)* 498 to 499 labelled a_1(0)* 498 to 500 labelled b_1(1)* 499 to 490 labelled a_1(0)* 499 to 500 labelled b_1(1)* 500 to 501 labelled b_1(1)* 501 to 502 labelled a_1(1)* 502 to 503 labelled a_1(1)* 503 to 504 labelled b_1(1)* 504 to 505 labelled b_1(1)* 505 to 506 labelled a_1(1)* 505 to 500 labelled b_1(1)* 506 to 507 labelled a_1(1)* 506 to 500 labelled b_1(1)* 507 to 508 labelled a_1(1)* 507 to 500 labelled b_1(1)* 508 to 490 labelled a_1(1)* 508 to 500 labelled b_1(1)


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(6)
YES