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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 4 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(x1)))))))) -> a(a(a(a(a(a(a(b(a(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(b(a(x1)))))))) -> b(a(a(a(b(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(b(a(x1)))))))) -> b(a(a(a(b(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(b(x))))))) -> b(a(a(a(b(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(b(x))))))) -> b(a(a(a(b(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(b(x))))))) -> b(a(a(a(b(a(a(a(a(a(a(x)))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353

Node 332 is start node and node 333 is final node.

Those nodes are connected through the following edges:

* 332 to 334 labelled b_1(0)* 333 to 333 labelled #_1(0)* 334 to 335 labelled a_1(0)* 335 to 336 labelled a_1(0)* 336 to 337 labelled a_1(0)* 337 to 338 labelled b_1(0)* 338 to 339 labelled a_1(0)* 338 to 344 labelled b_1(1)* 339 to 340 labelled a_1(0)* 339 to 344 labelled b_1(1)* 340 to 341 labelled a_1(0)* 340 to 344 labelled b_1(1)* 341 to 342 labelled a_1(0)* 341 to 344 labelled b_1(1)* 342 to 343 labelled a_1(0)* 342 to 344 labelled b_1(1)* 343 to 333 labelled a_1(0)* 343 to 344 labelled b_1(1)* 344 to 345 labelled a_1(1)* 345 to 346 labelled a_1(1)* 346 to 347 labelled a_1(1)* 347 to 348 labelled b_1(1)* 348 to 349 labelled a_1(1)* 348 to 344 labelled b_1(1)* 349 to 350 labelled a_1(1)* 349 to 344 labelled b_1(1)* 350 to 351 labelled a_1(1)* 350 to 344 labelled b_1(1)* 351 to 352 labelled a_1(1)* 351 to 344 labelled b_1(1)* 352 to 353 labelled a_1(1)* 352 to 344 labelled b_1(1)* 353 to 333 labelled a_1(1)* 353 to 344 labelled b_1(1)


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(6)
YES