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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170

Node 84 is start node and node 85 is final node.

Those nodes are connected through the following edges:

* 84 to 86 labelled b_1(0)* 85 to 85 labelled #_1(0)* 86 to 87 labelled a_1(0)* 87 to 88 labelled b_1(0)* 88 to 89 labelled a_1(0)* 89 to 90 labelled a_1(0)* 90 to 91 labelled a_1(0)* 91 to 92 labelled b_1(0)* 92 to 93 labelled a_1(0)* 92 to 159 labelled b_1(1)* 93 to 94 labelled a_1(0)* 93 to 159 labelled b_1(1)* 94 to 95 labelled a_1(0)* 94 to 159 labelled b_1(1)* 95 to 96 labelled a_1(0)* 95 to 159 labelled b_1(1)* 96 to 97 labelled a_1(0)* 96 to 159 labelled b_1(1)* 97 to 85 labelled a_1(0)* 97 to 159 labelled b_1(1)* 159 to 160 labelled a_1(1)* 160 to 161 labelled b_1(1)* 161 to 162 labelled a_1(1)* 162 to 163 labelled a_1(1)* 163 to 164 labelled a_1(1)* 164 to 165 labelled b_1(1)* 165 to 166 labelled a_1(1)* 165 to 159 labelled b_1(1)* 166 to 167 labelled a_1(1)* 166 to 159 labelled b_1(1)* 167 to 168 labelled a_1(1)* 167 to 159 labelled b_1(1)* 168 to 169 labelled a_1(1)* 168 to 159 labelled b_1(1)* 169 to 170 labelled a_1(1)* 169 to 159 labelled b_1(1)* 170 to 85 labelled a_1(1)* 170 to 159 labelled b_1(1)


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(6)
YES