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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) Strip Symbols Proof [SOUND, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(4) YES


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

   b(a(a(b(b(a(a(a(a(b(x1)))))))))) -> a(a(b(b(a(a(a(a(b(b(a(a(b(x1)))))))))))))

Q is empty.

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(1) Strip Symbols Proof (SOUND)
We were given the following TRS:

   b(a(a(b(b(a(a(a(a(b(x1)))))))))) -> a(a(b(b(a(a(a(a(b(b(a(a(b(x1)))))))))))))

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]: 

   b(a(a(b(b(a(a(a(a(x))))))))) -> a(a(b(b(a(a(a(a(b(b(a(a(x))))))))))))

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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(a(a(b(b(a(a(a(a(x))))))))) -> a(a(b(b(a(a(a(a(b(b(a(a(x))))))))))))

Q is empty.

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(3) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

   b(a(a(b(b(a(a(a(a(x))))))))) -> a(a(b(b(a(a(a(a(b(b(a(a(x))))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61

Node 2 is start node and node 5 is final node.

Those nodes are connected through the following edges:

* 2 to 6 labelled a_1(0)* 5 to 5 labelled #_1(0)* 6 to 7 labelled a_1(0)* 7 to 8 labelled b_1(0)* 8 to 9 labelled b_1(0)* 9 to 10 labelled a_1(0)* 10 to 11 labelled a_1(0)* 11 to 12 labelled a_1(0)* 12 to 13 labelled a_1(0)* 13 to 14 labelled b_1(0)* 13 to 40 labelled a_1(1)* 14 to 15 labelled b_1(0)* 14 to 29 labelled a_1(1)* 15 to 16 labelled a_1(0)* 16 to 5 labelled a_1(0)* 29 to 30 labelled a_1(1)* 30 to 31 labelled b_1(1)* 31 to 32 labelled b_1(1)* 32 to 33 labelled a_1(1)* 33 to 34 labelled a_1(1)* 34 to 35 labelled a_1(1)* 35 to 36 labelled a_1(1)* 36 to 37 labelled b_1(1)* 36 to 51 labelled a_1(2)* 37 to 38 labelled b_1(1)* 37 to 29 labelled a_1(1)* 38 to 39 labelled a_1(1)* 39 to 5 labelled a_1(1)* 40 to 41 labelled a_1(1)* 41 to 42 labelled b_1(1)* 42 to 43 labelled b_1(1)* 43 to 44 labelled a_1(1)* 44 to 45 labelled a_1(1)* 45 to 46 labelled a_1(1)* 46 to 47 labelled a_1(1)* 47 to 48 labelled b_1(1)* 47 to 51 labelled a_1(2)* 48 to 49 labelled b_1(1)* 48 to 29 labelled a_1(1)* 49 to 50 labelled a_1(1)* 50 to 36 labelled a_1(1)* 51 to 52 labelled a_1(2)* 52 to 53 labelled b_1(2)* 53 to 54 labelled b_1(2)* 54 to 55 labelled a_1(2)* 55 to 56 labelled a_1(2)* 56 to 57 labelled a_1(2)* 57 to 58 labelled a_1(2)* 58 to 59 labelled b_1(2)* 58 to 51 labelled a_1(2)* 59 to 60 labelled b_1(2)* 59 to 29 labelled a_1(1)* 60 to 61 labelled a_1(2)* 61 to 36 labelled a_1(2)


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(4)
YES