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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) Strip Symbols Proof [SOUND, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(a(a(b(a(b(a(a(a(b(x1)))))))))) -> a(b(a(b(a(a(a(b(a(b(a(a(b(x1)))))))))))))

Q is empty.

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(1) Strip Symbols Proof (SOUND)
We were given the following TRS:

   b(a(a(b(a(b(a(a(a(b(x1)))))))))) -> a(b(a(b(a(a(a(b(a(b(a(a(b(x1)))))))))))))

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]: 

   b(a(a(b(a(b(a(a(a(x))))))))) -> a(b(a(b(a(a(a(b(a(b(a(a(x))))))))))))

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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(a(a(b(a(b(a(a(a(x))))))))) -> a(b(a(b(a(a(a(b(a(b(a(a(x))))))))))))

Q is empty.

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(3) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

   b(a(a(b(a(b(a(a(a(x))))))))) -> a(b(a(b(a(a(a(b(a(b(a(a(x))))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216

Node 171 is start node and node 172 is final node.

Those nodes are connected through the following edges:

* 171 to 173 labelled a_1(0)* 172 to 172 labelled #_1(0)* 173 to 174 labelled b_1(0)* 174 to 175 labelled a_1(0)* 175 to 176 labelled b_1(0)* 176 to 177 labelled a_1(0)* 177 to 178 labelled a_1(0)* 178 to 179 labelled a_1(0)* 179 to 180 labelled b_1(0)* 179 to 195 labelled a_1(1)* 180 to 181 labelled a_1(0)* 181 to 182 labelled b_1(0)* 181 to 184 labelled a_1(1)* 182 to 183 labelled a_1(0)* 183 to 172 labelled a_1(0)* 184 to 185 labelled b_1(1)* 185 to 186 labelled a_1(1)* 186 to 187 labelled b_1(1)* 187 to 188 labelled a_1(1)* 188 to 189 labelled a_1(1)* 189 to 190 labelled a_1(1)* 190 to 191 labelled b_1(1)* 190 to 206 labelled a_1(2)* 191 to 192 labelled a_1(1)* 192 to 193 labelled b_1(1)* 192 to 184 labelled a_1(1)* 193 to 194 labelled a_1(1)* 194 to 172 labelled a_1(1)* 195 to 196 labelled b_1(1)* 196 to 197 labelled a_1(1)* 197 to 198 labelled b_1(1)* 198 to 199 labelled a_1(1)* 199 to 200 labelled a_1(1)* 200 to 201 labelled a_1(1)* 201 to 202 labelled b_1(1)* 201 to 206 labelled a_1(2)* 202 to 203 labelled a_1(1)* 203 to 204 labelled b_1(1)* 203 to 184 labelled a_1(1)* 204 to 205 labelled a_1(1)* 205 to 190 labelled a_1(1)* 206 to 207 labelled b_1(2)* 207 to 208 labelled a_1(2)* 208 to 209 labelled b_1(2)* 209 to 210 labelled a_1(2)* 210 to 211 labelled a_1(2)* 211 to 212 labelled a_1(2)* 212 to 213 labelled b_1(2)* 212 to 206 labelled a_1(2)* 213 to 214 labelled a_1(2)* 214 to 215 labelled b_1(2)* 214 to 184 labelled a_1(1)* 215 to 216 labelled a_1(2)* 216 to 190 labelled a_1(2)


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(4)
YES