YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(a(x1))))))))) -> a(a(a(a(a(a(b(b(b(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(a(b(b(b(a(x1))))))))) -> b(b(b(a(a(b(b(b(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(b(b(b(a(x1))))))))) -> b(b(b(a(a(b(b(b(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(b(b(b(x)))))))) -> b(b(b(a(a(b(b(b(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(b(b(b(x)))))))) -> b(b(b(a(a(b(b(b(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(b(b(b(x)))))))) -> b(b(b(a(a(b(b(b(a(a(a(a(a(x)))))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148

Node 123 is start node and node 124 is final node.

Those nodes are connected through the following edges:

* 123 to 125 labelled b_1(0)* 124 to 124 labelled #_1(0)* 125 to 126 labelled b_1(0)* 126 to 127 labelled b_1(0)* 127 to 128 labelled a_1(0)* 128 to 129 labelled a_1(0)* 129 to 130 labelled b_1(0)* 130 to 131 labelled b_1(0)* 131 to 132 labelled b_1(0)* 132 to 133 labelled a_1(0)* 132 to 137 labelled b_1(1)* 133 to 134 labelled a_1(0)* 133 to 137 labelled b_1(1)* 134 to 135 labelled a_1(0)* 134 to 137 labelled b_1(1)* 135 to 136 labelled a_1(0)* 135 to 137 labelled b_1(1)* 136 to 124 labelled a_1(0)* 136 to 137 labelled b_1(1)* 137 to 138 labelled b_1(1)* 138 to 139 labelled b_1(1)* 139 to 140 labelled a_1(1)* 140 to 141 labelled a_1(1)* 141 to 142 labelled b_1(1)* 142 to 143 labelled b_1(1)* 143 to 144 labelled b_1(1)* 144 to 145 labelled a_1(1)* 144 to 137 labelled b_1(1)* 145 to 146 labelled a_1(1)* 145 to 137 labelled b_1(1)* 146 to 147 labelled a_1(1)* 146 to 137 labelled b_1(1)* 147 to 148 labelled a_1(1)* 147 to 137 labelled b_1(1)* 148 to 124 labelled a_1(1)* 148 to 137 labelled b_1(1)


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(6)
YES