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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 1 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(b(a(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(a(b(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(a(b(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(a(b(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(a(b(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(a(b(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:
369, 370, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459

Node 369 is start node and node 370 is final node.

Those nodes are connected through the following edges:

* 369 to 436 labelled b_1(0)* 370 to 370 labelled #_1(0)* 436 to 437 labelled a_1(0)* 437 to 438 labelled a_1(0)* 438 to 439 labelled b_1(0)* 439 to 440 labelled a_1(0)* 440 to 441 labelled b_1(0)* 441 to 442 labelled a_1(0)* 441 to 448 labelled b_1(1)* 442 to 443 labelled a_1(0)* 442 to 448 labelled b_1(1)* 443 to 444 labelled a_1(0)* 443 to 448 labelled b_1(1)* 444 to 445 labelled a_1(0)* 444 to 448 labelled b_1(1)* 445 to 446 labelled a_1(0)* 445 to 448 labelled b_1(1)* 446 to 447 labelled a_1(0)* 446 to 448 labelled b_1(1)* 447 to 370 labelled a_1(0)* 447 to 448 labelled b_1(1)* 448 to 449 labelled a_1(1)* 449 to 450 labelled a_1(1)* 450 to 451 labelled b_1(1)* 451 to 452 labelled a_1(1)* 452 to 453 labelled b_1(1)* 453 to 454 labelled a_1(1)* 453 to 448 labelled b_1(1)* 454 to 455 labelled a_1(1)* 454 to 448 labelled b_1(1)* 455 to 456 labelled a_1(1)* 455 to 448 labelled b_1(1)* 456 to 457 labelled a_1(1)* 456 to 448 labelled b_1(1)* 457 to 458 labelled a_1(1)* 457 to 448 labelled b_1(1)* 458 to 459 labelled a_1(1)* 458 to 448 labelled b_1(1)* 459 to 370 labelled a_1(1)* 459 to 448 labelled b_1(1)


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(6)
YES