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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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

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

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

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

Those nodes are connected through the following edges:

* 1 to 2 labelled d_1(0), a_1(0), d_1(1), a_1(1)* 1 to 3 labelled a_1(0)* 1 to 4 labelled f_1(0)* 1 to 1 labelled f_1(0)* 1 to 8 labelled f_1(1), a_1(1), a_1(2)* 2 to 2 labelled #_1(0)* 3 to 2 labelled b_1(0), d_1(2)* 3 to 9 labelled a_1(1)* 3 to 10 labelled d_1(2)* 3 to 8 labelled f_1(1), a_1(2)* 4 to 5 labelled a_1(0)* 4 to 6 labelled d_1(1)* 5 to 6 labelled b_1(0)* 6 to 7 labelled b_1(0)* 7 to 2 labelled c_1(0)* 8 to 9 labelled a_1(1)* 8 to 10 labelled d_1(2)* 8 to 2 labelled d_1(2)* 8 to 8 labelled f_1(1), a_1(2)* 9 to 10 labelled b_1(1), d_1(2)* 9 to 2 labelled b_1(1), d_1(2)* 9 to 9 labelled a_1(1)* 9 to 8 labelled f_1(1), a_1(2)* 10 to 11 labelled b_1(1)* 11 to 2 labelled c_1(1)


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(2)
YES