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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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

   a(l(x1)) -> l(a(x1))
   r(a(a(x1))) -> a(a(r(x1)))
   b(l(x1)) -> b(a(r(x1)))
   r(b(x1)) -> l(b(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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

   a(l(x1)) -> l(a(x1))
   r(a(a(x1))) -> a(a(r(x1)))
   b(l(x1)) -> b(a(r(x1)))
   r(b(x1)) -> l(b(x1))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 3, 8, 10, 11, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30

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

Those nodes are connected through the following edges:

* 1 to 8 labelled l_1(0)* 1 to 10 labelled a_1(0), b_1(0)* 1 to 22 labelled l_1(1)* 1 to 23 labelled b_1(1)* 3 to 3 labelled #_1(0)* 8 to 3 labelled a_1(0), b_1(0)* 8 to 16 labelled l_1(1)* 8 to 17 labelled b_1(1)* 8 to 26 labelled b_1(2)* 10 to 11 labelled a_1(0)* 10 to 21 labelled l_1(1)* 11 to 3 labelled r_1(0)* 11 to 17 labelled a_1(1)* 11 to 20 labelled l_1(1)* 11 to 28 labelled l_1(2)* 16 to 3 labelled a_1(1)* 16 to 16 labelled l_1(1)* 17 to 18 labelled a_1(1)* 17 to 25 labelled l_1(2)* 18 to 3 labelled r_1(1)* 18 to 17 labelled a_1(1)* 18 to 20 labelled l_1(1)* 18 to 28 labelled l_1(2)* 20 to 3 labelled b_1(1)* 20 to 17 labelled b_1(1)* 20 to 26 labelled b_1(2)* 21 to 20 labelled a_1(1)* 21 to 28 labelled a_1(1)* 22 to 21 labelled a_1(1)* 23 to 24 labelled a_1(1)* 24 to 21 labelled r_1(1)* 24 to 26 labelled a_1(2)* 25 to 20 labelled a_1(2)* 25 to 28 labelled a_1(2)* 26 to 27 labelled a_1(2)* 27 to 25 labelled r_1(2)* 27 to 29 labelled a_1(3)* 28 to 25 labelled a_1(2)* 29 to 30 labelled a_1(3)* 30 to 25 labelled r_1(3)* 30 to 29 labelled a_1(3)


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(2)
YES