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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 1 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(x1) -> c(a(c(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)))
   b(x1) -> c(a(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)))
   b(x1) -> c(a(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:
2, 4, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42

Node 2 is start node and node 4 is final node.

Those nodes are connected through the following edges:

* 2 to 17 labelled a_1(0)* 2 to 19 labelled c_1(0)* 2 to 21 labelled a_1(0)* 2 to 25 labelled a_1(1)* 4 to 4 labelled #_1(0)* 17 to 18 labelled a_1(0)* 17 to 33 labelled a_1(1)* 18 to 4 labelled b_1(0)* 18 to 23 labelled c_1(1)* 18 to 27 labelled a_1(1)* 18 to 37 labelled a_1(2)* 19 to 20 labelled a_1(0)* 20 to 4 labelled c_1(0)* 21 to 22 labelled c_1(0)* 22 to 4 labelled a_1(0)* 22 to 29 labelled a_1(1)* 23 to 24 labelled a_1(1)* 24 to 4 labelled c_1(1)* 25 to 26 labelled c_1(1)* 26 to 18 labelled a_1(1)* 26 to 35 labelled a_1(2)* 26 to 33 labelled a_1(1)* 27 to 28 labelled a_1(1)* 27 to 35 labelled a_1(2)* 28 to 4 labelled b_1(1)* 28 to 31 labelled c_1(2)* 28 to 27 labelled a_1(1)* 28 to 37 labelled a_1(2)* 29 to 30 labelled c_1(1)* 30 to 4 labelled a_1(1)* 30 to 29 labelled a_1(1)* 31 to 32 labelled a_1(2)* 32 to 4 labelled c_1(2)* 33 to 34 labelled c_1(1)* 34 to 27 labelled a_1(1)* 34 to 37 labelled a_1(1), a_1(2)* 35 to 36 labelled c_1(2)* 36 to 27 labelled a_1(2)* 36 to 37 labelled a_1(2)* 36 to 39 labelled a_1(3)* 37 to 38 labelled c_1(2)* 38 to 28 labelled a_1(2)* 38 to 35 labelled a_1(2)* 38 to 41 labelled a_1(3)* 39 to 40 labelled c_1(3)* 40 to 35 labelled a_1(3)* 41 to 42 labelled c_1(3)* 42 to 37 labelled a_1(3)


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(4)
YES