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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 1 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(x1)))))))) -> a(a(a(a(a(a(b(b(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(b(b(a(x1)))))))) -> b(b(a(b(b(a(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(a(x1)))))))) -> b(b(a(b(b(a(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(x))))))) -> b(b(a(b(b(a(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(x))))))) -> b(b(a(b(b(a(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(x))))))) -> b(b(a(b(b(a(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:
7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Node 7 is start node and node 8 is final node.

Those nodes are connected through the following edges:

* 7 to 35 labelled b_1(0)* 8 to 8 labelled #_1(0)* 35 to 36 labelled b_1(0)* 36 to 37 labelled a_1(0)* 37 to 38 labelled b_1(0)* 38 to 39 labelled b_1(0)* 39 to 40 labelled a_1(0)* 40 to 41 labelled b_1(0)* 41 to 42 labelled b_1(0)* 42 to 43 labelled a_1(0)* 42 to 59 labelled b_1(1)* 43 to 44 labelled a_1(0)* 43 to 59 labelled b_1(1)* 44 to 45 labelled a_1(0)* 44 to 59 labelled b_1(1)* 45 to 46 labelled a_1(0)* 45 to 59 labelled b_1(1)* 46 to 8 labelled a_1(0)* 46 to 59 labelled b_1(1)* 59 to 60 labelled b_1(1)* 60 to 61 labelled a_1(1)* 61 to 62 labelled b_1(1)* 62 to 63 labelled b_1(1)* 63 to 64 labelled a_1(1)* 64 to 65 labelled b_1(1)* 65 to 66 labelled b_1(1)* 66 to 67 labelled a_1(1)* 66 to 59 labelled b_1(1)* 67 to 68 labelled a_1(1)* 67 to 59 labelled b_1(1)* 68 to 69 labelled a_1(1)* 68 to 59 labelled b_1(1)* 69 to 70 labelled a_1(1)* 69 to 59 labelled b_1(1)* 70 to 8 labelled a_1(1)* 70 to 59 labelled b_1(1)


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(6)
YES