/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(4) YES


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

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

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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 50, 51, 52, 53, 54

Node 2 is start node and node 3 is final node.

Those nodes are connected through the following edges:

* 2 to 5 labelled a_1(0)* 2 to 9 labelled a_1(0)* 3 to 3 labelled #_1(0)* 5 to 6 labelled b_1(0)* 6 to 7 labelled a_1(0)* 7 to 8 labelled b_1(0)* 7 to 27 labelled a_1(1)* 8 to 3 labelled c_1(0)* 9 to 10 labelled b_1(0)* 10 to 11 labelled c_1(0)* 11 to 12 labelled b_1(0)* 12 to 13 labelled a_1(0)* 12 to 23 labelled a_1(1)* 13 to 3 labelled a_1(0)* 13 to 23 labelled a_1(1)* 23 to 24 labelled b_1(1)* 24 to 25 labelled a_1(1)* 25 to 26 labelled b_1(1)* 25 to 27 labelled a_1(1)* 25 to 32 labelled a_1(2)* 26 to 3 labelled c_1(1)* 26 to 25 labelled c_1(1)* 27 to 28 labelled b_1(1)* 28 to 29 labelled c_1(1)* 29 to 30 labelled b_1(1)* 30 to 31 labelled a_1(1)* 30 to 23 labelled a_1(1)* 30 to 37 labelled a_1(2)* 31 to 3 labelled a_1(1)* 31 to 23 labelled a_1(1)* 32 to 33 labelled b_1(2)* 33 to 34 labelled c_1(2)* 34 to 35 labelled b_1(2)* 35 to 36 labelled a_1(2)* 36 to 27 labelled a_1(2)* 36 to 32 labelled a_1(2)* 37 to 38 labelled b_1(2)* 38 to 39 labelled a_1(2)* 39 to 40 labelled b_1(2)* 39 to 32 labelled a_1(2)* 39 to 50 labelled a_1(3)* 40 to 25 labelled c_1(2)* 50 to 51 labelled b_1(3)* 51 to 52 labelled c_1(3)* 52 to 53 labelled b_1(3)* 53 to 54 labelled a_1(3)* 54 to 32 labelled a_1(3)


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(4)
YES