YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(x1)))))) -> a(a(a(a(a(b(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(b(a(x1)))))) -> b(a(b(a(a(b(a(a(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(a(x1)))))) -> b(a(b(a(a(b(a(a(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(x))))) -> b(a(b(a(a(b(a(a(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(x))))) -> b(a(b(a(a(b(a(a(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(x))))) -> b(a(b(a(a(b(a(a(b(a(a(a(a(x)))))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 425, 426, 428, 430, 432, 434, 436, 438, 440, 442, 444, 446

Node 385 is start node and node 386 is final node.

Those nodes are connected through the following edges:

* 385 to 387 labelled b_1(0)* 386 to 386 labelled #_1(0)* 387 to 388 labelled a_1(0)* 388 to 389 labelled b_1(0)* 389 to 390 labelled a_1(0)* 390 to 391 labelled a_1(0)* 391 to 392 labelled b_1(0)* 392 to 393 labelled a_1(0)* 393 to 394 labelled a_1(0)* 394 to 395 labelled b_1(0)* 395 to 396 labelled a_1(0)* 395 to 425 labelled b_1(1)* 396 to 397 labelled a_1(0)* 396 to 425 labelled b_1(1)* 397 to 398 labelled a_1(0)* 397 to 425 labelled b_1(1)* 398 to 386 labelled a_1(0)* 398 to 425 labelled b_1(1)* 425 to 426 labelled a_1(1)* 426 to 428 labelled b_1(1)* 428 to 430 labelled a_1(1)* 430 to 432 labelled a_1(1)* 432 to 434 labelled b_1(1)* 434 to 436 labelled a_1(1)* 436 to 438 labelled a_1(1)* 438 to 440 labelled b_1(1)* 440 to 442 labelled a_1(1)* 440 to 425 labelled b_1(1)* 442 to 444 labelled a_1(1)* 442 to 425 labelled b_1(1)* 444 to 446 labelled a_1(1)* 444 to 425 labelled b_1(1)* 446 to 386 labelled a_1(1)* 446 to 425 labelled b_1(1)


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(6)
YES