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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(a(b(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(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(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(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(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(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:
262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285

Node 262 is start node and node 263 is final node.

Those nodes are connected through the following edges:

* 262 to 264 labelled b_1(0)* 263 to 263 labelled #_1(0)* 264 to 265 labelled a_1(0)* 265 to 266 labelled b_1(0)* 266 to 267 labelled a_1(0)* 267 to 268 labelled b_1(0)* 268 to 269 labelled a_1(0)* 269 to 270 labelled a_1(0)* 270 to 271 labelled b_1(0)* 271 to 272 labelled a_1(0)* 271 to 275 labelled b_1(1)* 272 to 273 labelled a_1(0)* 272 to 275 labelled b_1(1)* 273 to 274 labelled a_1(0)* 273 to 275 labelled b_1(1)* 274 to 263 labelled a_1(0)* 274 to 275 labelled b_1(1)* 275 to 276 labelled a_1(1)* 276 to 277 labelled b_1(1)* 277 to 278 labelled a_1(1)* 278 to 279 labelled b_1(1)* 279 to 280 labelled a_1(1)* 280 to 281 labelled a_1(1)* 281 to 282 labelled b_1(1)* 282 to 283 labelled a_1(1)* 282 to 275 labelled b_1(1)* 283 to 284 labelled a_1(1)* 283 to 275 labelled b_1(1)* 284 to 285 labelled a_1(1)* 284 to 275 labelled b_1(1)* 285 to 263 labelled a_1(1)* 285 to 275 labelled b_1(1)


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(6)
YES