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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237

Node 218 is start node and node 219 is final node.

Those nodes are connected through the following edges:

* 218 to 220 labelled b_1(0)* 219 to 219 labelled #_1(0)* 220 to 221 labelled a_1(0)* 221 to 222 labelled b_1(0)* 222 to 223 labelled a_1(0)* 223 to 224 labelled b_1(0)* 224 to 225 labelled a_1(0)* 224 to 229 labelled b_1(1)* 225 to 226 labelled a_1(0)* 225 to 229 labelled b_1(1)* 226 to 227 labelled a_1(0)* 226 to 229 labelled b_1(1)* 227 to 228 labelled a_1(0)* 227 to 229 labelled b_1(1)* 228 to 219 labelled a_1(0)* 228 to 229 labelled b_1(1)* 229 to 230 labelled a_1(1)* 230 to 231 labelled b_1(1)* 231 to 232 labelled a_1(1)* 232 to 233 labelled b_1(1)* 233 to 234 labelled a_1(1)* 233 to 229 labelled b_1(1)* 234 to 235 labelled a_1(1)* 234 to 229 labelled b_1(1)* 235 to 236 labelled a_1(1)* 235 to 229 labelled b_1(1)* 236 to 237 labelled a_1(1)* 236 to 229 labelled b_1(1)* 237 to 219 labelled a_1(1)* 237 to 229 labelled b_1(1)


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(6)
YES