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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
344, 345, 346, 347, 348, 349, 351, 352, 354, 356, 358, 360, 363, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391

Node 344 is start node and node 345 is final node.

Those nodes are connected through the following edges:

* 344 to 346 labelled b_1(0)* 345 to 345 labelled #_1(0)* 346 to 347 labelled a_1(0)* 347 to 348 labelled a_1(0)* 348 to 349 labelled b_1(0)* 349 to 351 labelled a_1(0)* 351 to 352 labelled b_1(0)* 352 to 354 labelled a_1(0)* 354 to 356 labelled b_1(0)* 356 to 358 labelled a_1(0)* 356 to 381 labelled b_1(1)* 358 to 360 labelled a_1(0)* 358 to 381 labelled b_1(1)* 360 to 363 labelled a_1(0)* 360 to 381 labelled b_1(1)* 363 to 345 labelled a_1(0)* 363 to 381 labelled b_1(1)* 381 to 382 labelled a_1(1)* 382 to 383 labelled a_1(1)* 383 to 384 labelled b_1(1)* 384 to 385 labelled a_1(1)* 385 to 386 labelled b_1(1)* 386 to 387 labelled a_1(1)* 387 to 388 labelled b_1(1)* 388 to 389 labelled a_1(1)* 388 to 381 labelled b_1(1)* 389 to 390 labelled a_1(1)* 389 to 381 labelled b_1(1)* 390 to 391 labelled a_1(1)* 390 to 381 labelled b_1(1)* 391 to 345 labelled a_1(1)* 391 to 381 labelled b_1(1)


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(6)
YES