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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(a(a(a(a(a(a(a(x1))))))))) -> a(a(a(a(a(a(a(a(b(a(a(b(a(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(a(a(a(b(a(x1))))))))) -> b(a(b(a(a(b(a(a(a(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(a(a(a(b(a(x1))))))))) -> b(a(b(a(a(b(a(a(a(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(a(a(a(b(x)))))))) -> b(a(b(a(a(b(a(a(a(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(a(a(a(b(x)))))))) -> b(a(b(a(a(b(a(a(a(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(a(a(a(b(x)))))))) -> b(a(b(a(a(b(a(a(a(a(a(a(a(x)))))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
447, 448, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498

Node 447 is start node and node 448 is final node.

Those nodes are connected through the following edges:

* 447 to 475 labelled b_1(0)* 448 to 448 labelled #_1(0)* 475 to 476 labelled a_1(0)* 476 to 477 labelled b_1(0)* 477 to 478 labelled a_1(0)* 478 to 479 labelled a_1(0)* 479 to 480 labelled b_1(0)* 480 to 481 labelled a_1(0)* 480 to 487 labelled b_1(1)* 481 to 482 labelled a_1(0)* 481 to 487 labelled b_1(1)* 482 to 483 labelled a_1(0)* 482 to 487 labelled b_1(1)* 483 to 484 labelled a_1(0)* 483 to 487 labelled b_1(1)* 484 to 485 labelled a_1(0)* 484 to 487 labelled b_1(1)* 485 to 486 labelled a_1(0)* 485 to 487 labelled b_1(1)* 486 to 448 labelled a_1(0)* 486 to 487 labelled b_1(1)* 487 to 488 labelled a_1(1)* 488 to 489 labelled b_1(1)* 489 to 490 labelled a_1(1)* 490 to 491 labelled a_1(1)* 491 to 492 labelled b_1(1)* 492 to 493 labelled a_1(1)* 492 to 487 labelled b_1(1)* 493 to 494 labelled a_1(1)* 493 to 487 labelled b_1(1)* 494 to 495 labelled a_1(1)* 494 to 487 labelled b_1(1)* 495 to 496 labelled a_1(1)* 495 to 487 labelled b_1(1)* 496 to 497 labelled a_1(1)* 496 to 487 labelled b_1(1)* 497 to 498 labelled a_1(1)* 497 to 487 labelled b_1(1)* 498 to 448 labelled a_1(1)* 498 to 487 labelled b_1(1)


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(6)
YES