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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 8 ms]
(6) YES


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

   a(b(b(a(a(a(a(a(a(a(x1)))))))))) -> a(a(a(a(a(a(a(a(b(b(a(b(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(b(a(x1)))))))))) -> b(b(a(b(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(b(a(x1)))))))))) -> b(b(a(b(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(b(x))))))))) -> b(b(a(b(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(b(x))))))))) -> b(b(a(b(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(b(x))))))))) -> b(b(a(b(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:
77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Node 77 is start node and node 78 is final node.

Those nodes are connected through the following edges:

* 77 to 79 labelled b_1(0)* 78 to 78 labelled #_1(0)* 79 to 80 labelled b_1(0)* 80 to 81 labelled a_1(0)* 81 to 82 labelled b_1(0)* 82 to 83 labelled b_1(0)* 83 to 84 labelled a_1(0)* 83 to 114 labelled b_1(1)* 84 to 85 labelled a_1(0)* 84 to 114 labelled b_1(1)* 85 to 86 labelled a_1(0)* 85 to 114 labelled b_1(1)* 86 to 87 labelled a_1(0)* 86 to 114 labelled b_1(1)* 87 to 88 labelled a_1(0)* 87 to 114 labelled b_1(1)* 88 to 89 labelled a_1(0)* 88 to 114 labelled b_1(1)* 89 to 78 labelled a_1(0)* 89 to 114 labelled b_1(1)* 114 to 115 labelled b_1(1)* 115 to 116 labelled a_1(1)* 116 to 117 labelled b_1(1)* 117 to 118 labelled b_1(1)* 118 to 119 labelled a_1(1)* 118 to 114 labelled b_1(1)* 119 to 120 labelled a_1(1)* 119 to 114 labelled b_1(1)* 120 to 121 labelled a_1(1)* 120 to 114 labelled b_1(1)* 121 to 122 labelled a_1(1)* 121 to 114 labelled b_1(1)* 122 to 123 labelled a_1(1)* 122 to 114 labelled b_1(1)* 123 to 124 labelled a_1(1)* 123 to 114 labelled b_1(1)* 124 to 78 labelled a_1(1)* 124 to 114 labelled b_1(1)


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(6)
YES