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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(a(a(a(x1))))))))) -> a(a(a(a(a(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(a(a(a(b(a(x1))))))))) -> b(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(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(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(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(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:
162, 163, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199

Node 162 is start node and node 163 is final node.

Those nodes are connected through the following edges:

* 162 to 182 labelled b_1(0)* 163 to 163 labelled #_1(0)* 182 to 183 labelled a_1(0)* 183 to 184 labelled b_1(0)* 184 to 185 labelled a_1(0)* 184 to 191 labelled b_1(1)* 185 to 186 labelled a_1(0)* 185 to 191 labelled b_1(1)* 186 to 187 labelled a_1(0)* 186 to 191 labelled b_1(1)* 187 to 188 labelled a_1(0)* 187 to 191 labelled b_1(1)* 188 to 189 labelled a_1(0)* 188 to 191 labelled b_1(1)* 189 to 190 labelled a_1(0)* 189 to 191 labelled b_1(1)* 190 to 163 labelled a_1(0)* 190 to 191 labelled b_1(1)* 191 to 192 labelled a_1(1)* 192 to 193 labelled b_1(1)* 193 to 194 labelled a_1(1)* 193 to 191 labelled b_1(1)* 194 to 195 labelled a_1(1)* 194 to 191 labelled b_1(1)* 195 to 196 labelled a_1(1)* 195 to 191 labelled b_1(1)* 196 to 197 labelled a_1(1)* 196 to 191 labelled b_1(1)* 197 to 198 labelled a_1(1)* 197 to 191 labelled b_1(1)* 198 to 199 labelled a_1(1)* 198 to 191 labelled b_1(1)* 199 to 163 labelled a_1(1)* 199 to 191 labelled b_1(1)


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(6)
YES