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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 29 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))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(x1))) -> a(l(c(c(r(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))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(x1))) -> a(l(c(c(r(x1)))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27

Node 1 is start node and node 2 is final node.

Those nodes are connected through the following edges:

* 1 to 3 labelled l_1(0), c_1(0), r_1(0)* 1 to 4 labelled a_1(0)* 1 to 8 labelled l_1(1)* 1 to 9 labelled r_1(1)* 1 to 12 labelled a_1(1)* 1 to 16 labelled l_1(2)* 2 to 2 labelled #_1(0)* 3 to 2 labelled a_1(0)* 3 to 9 labelled l_1(1), c_1(1), r_1(1)* 3 to 17 labelled a_1(2)* 3 to 21 labelled l_1(3)* 4 to 5 labelled l_1(0)* 5 to 6 labelled c_1(0)* 6 to 7 labelled c_1(0)* 7 to 2 labelled r_1(0)* 8 to 5 labelled a_1(1)* 8 to 10 labelled c_1(1)* 9 to 2 labelled a_1(1)* 9 to 9 labelled l_1(1), c_1(1), r_1(1)* 9 to 17 labelled a_1(2)* 9 to 21 labelled l_1(3)* 10 to 6 labelled a_1(1)* 10 to 11 labelled c_1(1)* 10 to 9 labelled r_1(1)* 11 to 7 labelled a_1(1)* 12 to 13 labelled l_1(1)* 13 to 14 labelled c_1(1)* 14 to 15 labelled c_1(1)* 15 to 2 labelled r_1(1)* 15 to 17 labelled r_1(1)* 16 to 13 labelled a_1(2)* 16 to 22 labelled c_1(2)* 17 to 18 labelled l_1(2)* 18 to 19 labelled c_1(2)* 19 to 20 labelled c_1(2)* 20 to 2 labelled r_1(2)* 20 to 17 labelled r_1(2)* 21 to 18 labelled a_1(3)* 21 to 23 labelled c_1(3)* 22 to 14 labelled a_1(2)* 22 to 24 labelled c_1(2)* 22 to 26 labelled r_1(2)* 22 to 3 labelled r_1(2)* 23 to 19 labelled a_1(3)* 23 to 25 labelled c_1(3)* 23 to 27 labelled r_1(3)* 24 to 15 labelled a_1(2)* 25 to 20 labelled a_1(3)* 26 to 2 labelled a_1(2)* 26 to 9 labelled l_1(1), c_1(1), r_1(1)* 26 to 17 labelled a_1(2)* 26 to 21 labelled l_1(3)* 27 to 2 labelled a_1(3)* 27 to 17 labelled a_1(3), a_1(2)* 27 to 9 labelled l_1(1), c_1(1), r_1(1)* 27 to 21 labelled l_1(3)


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(2)
YES