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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418

Node 397 is start node and node 398 is final node.

Those nodes are connected through the following edges:

* 397 to 399 labelled b_1(0)* 398 to 398 labelled #_1(0)* 399 to 400 labelled a_1(0)* 400 to 401 labelled b_1(0)* 401 to 402 labelled a_1(0)* 402 to 403 labelled a_1(0)* 403 to 404 labelled a_1(0)* 404 to 405 labelled b_1(0)* 405 to 406 labelled a_1(0)* 405 to 409 labelled b_1(1)* 406 to 407 labelled a_1(0)* 406 to 409 labelled b_1(1)* 407 to 408 labelled a_1(0)* 407 to 409 labelled b_1(1)* 408 to 398 labelled a_1(0)* 408 to 409 labelled b_1(1)* 409 to 410 labelled a_1(1)* 410 to 411 labelled b_1(1)* 411 to 412 labelled a_1(1)* 412 to 413 labelled a_1(1)* 413 to 414 labelled a_1(1)* 414 to 415 labelled b_1(1)* 415 to 416 labelled a_1(1)* 415 to 409 labelled b_1(1)* 416 to 417 labelled a_1(1)* 416 to 409 labelled b_1(1)* 417 to 418 labelled a_1(1)* 417 to 409 labelled b_1(1)* 418 to 398 labelled a_1(1)* 418 to 409 labelled b_1(1)


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(6)
YES