/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

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

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 43, 44, 45, 46

Node 13 is start node and node 14 is final node.

Those nodes are connected through the following edges:

* 13 to 15 labelled a_1(0)* 13 to 17 labelled b_1(0)* 13 to 20 labelled a_1(0)* 13 to 22 labelled a_1(1)* 14 to 14 labelled #_1(0)* 15 to 16 labelled a_1(0)* 15 to 31 labelled a_1(1)* 16 to 14 labelled b_1(0)* 16 to 24 labelled a_1(1)* 16 to 35 labelled a_1(2)* 17 to 14 labelled c_1(0)* 17 to 28 labelled b_1(1)* 20 to 21 labelled c_1(0)* 21 to 14 labelled a_1(0)* 21 to 29 labelled a_1(1)* 22 to 23 labelled c_1(1)* 23 to 16 labelled a_1(1)* 23 to 33 labelled a_1(2)* 23 to 31 labelled a_1(1)* 24 to 26 labelled a_1(1)* 24 to 33 labelled a_1(2)* 26 to 14 labelled b_1(1)* 26 to 24 labelled a_1(1)* 26 to 35 labelled a_1(2)* 28 to 14 labelled c_1(1)* 28 to 28 labelled b_1(1)* 29 to 30 labelled c_1(1)* 30 to 14 labelled a_1(1)* 30 to 29 labelled a_1(1)* 31 to 32 labelled c_1(1)* 32 to 24 labelled a_1(1)* 32 to 35 labelled a_1(1), a_1(2)* 33 to 34 labelled c_1(2)* 34 to 24 labelled a_1(2)* 34 to 35 labelled a_1(2)* 34 to 43 labelled a_1(3)* 35 to 36 labelled c_1(2)* 36 to 26 labelled a_1(2)* 36 to 33 labelled a_1(2)* 36 to 45 labelled a_1(3)* 43 to 44 labelled c_1(3)* 44 to 33 labelled a_1(3)* 45 to 46 labelled c_1(3)* 46 to 35 labelled a_1(3)


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(4)
YES