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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
70, 71, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121

Node 70 is start node and node 71 is final node.

Those nodes are connected through the following edges:

* 70 to 98 labelled b_1(0)* 71 to 71 labelled #_1(0)* 98 to 99 labelled b_1(0)* 99 to 100 labelled b_1(0)* 100 to 101 labelled a_1(0)* 101 to 102 labelled a_1(0)* 102 to 103 labelled a_1(0)* 103 to 104 labelled b_1(0)* 104 to 105 labelled b_1(0)* 105 to 106 labelled b_1(0)* 106 to 107 labelled a_1(0)* 106 to 110 labelled b_1(1)* 107 to 108 labelled a_1(0)* 107 to 110 labelled b_1(1)* 108 to 109 labelled a_1(0)* 108 to 110 labelled b_1(1)* 109 to 71 labelled a_1(0)* 109 to 110 labelled b_1(1)* 110 to 111 labelled b_1(1)* 111 to 112 labelled b_1(1)* 112 to 113 labelled a_1(1)* 113 to 114 labelled a_1(1)* 114 to 115 labelled a_1(1)* 115 to 116 labelled b_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 110 labelled b_1(1)* 119 to 120 labelled a_1(1)* 119 to 110 labelled b_1(1)* 120 to 121 labelled a_1(1)* 120 to 110 labelled b_1(1)* 121 to 71 labelled a_1(1)* 121 to 110 labelled b_1(1)


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(6)
YES