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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RFCMatchBoundsTRSProof [EQUIVALENT, 19 ms]
(2) YES


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

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

Q is empty.

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(1) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 3, 19, 20, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58

Node 1 is start node and node 3 is final node.

Those nodes are connected through the following edges:

* 1 to 19 labelled a_1(0)* 1 to 21 labelled a_1(0)* 1 to 31 labelled a_1(1)* 3 to 3 labelled #_1(0)* 19 to 20 labelled b_1(0)* 19 to 25 labelled a_1(1)* 20 to 3 labelled a_1(0)* 20 to 23 labelled a_1(1)* 20 to 33 labelled a_1(2)* 20 to 37 labelled a_1(3)* 21 to 22 labelled c_1(0)* 22 to 3 labelled a_1(0)* 22 to 23 labelled a_1(1)* 22 to 33 labelled a_1(2)* 22 to 37 labelled a_1(3)* 23 to 24 labelled b_1(1)* 23 to 25 labelled a_1(1)* 23 to 35 labelled a_1(2)* 24 to 3 labelled a_1(1)* 24 to 23 labelled a_1(1)* 24 to 33 labelled a_1(2)* 24 to 37 labelled a_1(3)* 25 to 26 labelled c_1(1)* 26 to 3 labelled a_1(1)* 26 to 23 labelled a_1(1)* 26 to 24 labelled a_1(1)* 26 to 33 labelled a_1(2)* 26 to 34 labelled a_1(1)* 26 to 37 labelled a_1(3)* 26 to 38 labelled a_1(1)* 31 to 32 labelled b_1(1)* 32 to 25 labelled a_1(1)* 33 to 34 labelled b_1(2)* 33 to 25 labelled a_1(1)* 33 to 35 labelled a_1(2)* 33 to 49 labelled a_1(3)* 34 to 25 labelled a_1(2)* 34 to 3 labelled a_1(2)* 34 to 23 labelled a_1(2), a_1(1)* 34 to 33 labelled a_1(2)* 34 to 35 labelled a_1(2)* 34 to 37 labelled a_1(3), a_1(2)* 34 to 49 labelled a_1(2)* 35 to 36 labelled c_1(2)* 36 to 24 labelled a_1(2)* 36 to 33 labelled a_1(2)* 36 to 39 labelled a_1(3)* 36 to 34 labelled a_1(2)* 36 to 37 labelled a_1(3)* 36 to 51 labelled a_1(4)* 36 to 38 labelled a_1(2)* 37 to 38 labelled b_1(3)* 38 to 35 labelled a_1(3)* 38 to 49 labelled a_1(3)* 39 to 40 labelled b_1(3)* 39 to 49 labelled a_1(3)* 39 to 25 labelled a_1(1)* 39 to 35 labelled a_1(2)* 39 to 55 labelled a_1(4)* 40 to 33 labelled a_1(3), a_1(2)* 40 to 37 labelled a_1(3)* 40 to 25 labelled a_1(3)* 40 to 3 labelled a_1(3)* 40 to 23 labelled a_1(3), a_1(1)* 40 to 51 labelled a_1(4)* 49 to 50 labelled c_1(3)* 50 to 34 labelled a_1(3)* 50 to 38 labelled a_1(3)* 50 to 39 labelled a_1(3)* 50 to 33 labelled a_1(2)* 50 to 37 labelled a_1(3)* 50 to 53 labelled a_1(4)* 50 to 51 labelled a_1(4)* 50 to 57 labelled a_1(5)* 51 to 52 labelled b_1(4)* 52 to 49 labelled a_1(4)* 52 to 55 labelled a_1(4)* 53 to 54 labelled b_1(4)* 53 to 55 labelled a_1(4)* 54 to 37 labelled a_1(4)* 54 to 35 labelled a_1(4)* 55 to 56 labelled c_1(4)* 56 to 38 labelled a_1(4)* 56 to 53 labelled a_1(4)* 56 to 51 labelled a_1(4)* 56 to 52 labelled a_1(4)* 56 to 57 labelled a_1(5)* 57 to 58 labelled b_1(5)* 58 to 55 labelled a_1(5)* 58 to 49 labelled a_1(5)


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(2)
YES