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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
294, 295, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326

Node 294 is start node and node 295 is final node.

Those nodes are connected through the following edges:

* 294 to 307 labelled b_1(0)* 295 to 295 labelled #_1(0)* 307 to 308 labelled a_1(0)* 308 to 309 labelled a_1(0)* 309 to 310 labelled b_1(0)* 310 to 311 labelled a_1(0)* 310 to 317 labelled b_1(1)* 311 to 312 labelled a_1(0)* 311 to 317 labelled b_1(1)* 312 to 313 labelled a_1(0)* 312 to 317 labelled b_1(1)* 313 to 314 labelled a_1(0)* 313 to 317 labelled b_1(1)* 314 to 315 labelled a_1(0)* 314 to 317 labelled b_1(1)* 315 to 316 labelled a_1(0)* 315 to 317 labelled b_1(1)* 316 to 295 labelled a_1(0)* 316 to 317 labelled b_1(1)* 317 to 318 labelled a_1(1)* 318 to 319 labelled a_1(1)* 319 to 320 labelled b_1(1)* 320 to 321 labelled a_1(1)* 320 to 317 labelled b_1(1)* 321 to 322 labelled a_1(1)* 321 to 317 labelled b_1(1)* 322 to 323 labelled a_1(1)* 322 to 317 labelled b_1(1)* 323 to 324 labelled a_1(1)* 323 to 317 labelled b_1(1)* 324 to 325 labelled a_1(1)* 324 to 317 labelled b_1(1)* 325 to 326 labelled a_1(1)* 325 to 317 labelled b_1(1)* 326 to 295 labelled a_1(1)* 326 to 317 labelled b_1(1)


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(6)
YES