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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159

Node 136 is start node and node 137 is final node.

Those nodes are connected through the following edges:

* 136 to 138 labelled b_1(0)* 137 to 137 labelled #_1(0)* 138 to 139 labelled a_1(0)* 139 to 140 labelled a_1(0)* 140 to 141 labelled a_1(0)* 141 to 142 labelled b_1(0)* 142 to 143 labelled a_1(0)* 143 to 144 labelled b_1(0)* 144 to 145 labelled a_1(0)* 144 to 149 labelled b_1(1)* 145 to 146 labelled a_1(0)* 145 to 149 labelled b_1(1)* 146 to 147 labelled a_1(0)* 146 to 149 labelled b_1(1)* 147 to 148 labelled a_1(0)* 147 to 149 labelled b_1(1)* 148 to 137 labelled a_1(0)* 148 to 149 labelled b_1(1)* 149 to 150 labelled a_1(1)* 150 to 151 labelled a_1(1)* 151 to 152 labelled a_1(1)* 152 to 153 labelled b_1(1)* 153 to 154 labelled a_1(1)* 154 to 155 labelled b_1(1)* 155 to 156 labelled a_1(1)* 155 to 149 labelled b_1(1)* 156 to 157 labelled a_1(1)* 156 to 149 labelled b_1(1)* 157 to 158 labelled a_1(1)* 157 to 149 labelled b_1(1)* 158 to 159 labelled a_1(1)* 158 to 149 labelled b_1(1)* 159 to 137 labelled a_1(1)* 159 to 149 labelled b_1(1)


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(6)
YES