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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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:

   b(c(a(x1))) -> a(b(a(b(x1))))
   b(x1) -> c(c(x1))
   a(a(x1)) -> a(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(c(b(x1))) -> b(a(b(a(x1))))
   b(x1) -> c(c(x1))
   a(a(x1)) -> a(b(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 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50

Node 30 is start node and node 31 is final node.

Those nodes are connected through the following edges:

* 30 to 32 labelled b_1(0)* 30 to 35 labelled c_1(0)* 30 to 36 labelled a_1(0)* 30 to 39 labelled c_1(1)* 31 to 31 labelled #_1(0)* 32 to 33 labelled a_1(0)* 33 to 34 labelled b_1(0)* 33 to 40 labelled c_1(1)* 34 to 31 labelled a_1(0)* 34 to 42 labelled b_1(1)* 34 to 45 labelled a_1(1)* 34 to 48 labelled c_1(2)* 35 to 31 labelled c_1(0)* 36 to 37 labelled b_1(0)* 36 to 41 labelled c_1(1)* 37 to 38 labelled c_1(0)* 38 to 31 labelled a_1(0)* 38 to 42 labelled b_1(1)* 38 to 45 labelled a_1(1)* 38 to 48 labelled c_1(2)* 39 to 32 labelled c_1(1)* 40 to 34 labelled c_1(1)* 41 to 37 labelled c_1(1)* 42 to 43 labelled a_1(1)* 43 to 44 labelled b_1(1)* 43 to 49 labelled c_1(2)* 44 to 31 labelled a_1(1)* 44 to 42 labelled b_1(1)* 44 to 45 labelled a_1(1)* 44 to 48 labelled c_1(2)* 45 to 46 labelled b_1(1)* 45 to 50 labelled c_1(2)* 46 to 47 labelled c_1(1)* 47 to 31 labelled a_1(1)* 47 to 42 labelled b_1(1)* 47 to 45 labelled a_1(1)* 47 to 48 labelled c_1(2)* 48 to 42 labelled c_1(2)* 49 to 44 labelled c_1(2)* 50 to 46 labelled c_1(2)


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(4)
YES