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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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__f(f(a)) -> c(f(g(f(a))))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(c(X)) -> c(X)
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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__f(f(a)) -> c(f(g(f(a))))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(c(X)) -> c(X)
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29

Node 14 is start node and node 15 is final node.

Those nodes are connected through the following edges:

* 14 to 16 labelled c_1(0)* 14 to 20 labelled a__f_1(0), g_1(0), f_1(1)* 14 to 15 labelled a(0), c_1(0), f_1(0)* 14 to 22 labelled c_1(1)* 15 to 15 labelled #_1(0)* 16 to 17 labelled f_1(0)* 17 to 18 labelled g_1(0)* 18 to 19 labelled f_1(0)* 19 to 15 labelled a(0)* 20 to 15 labelled mark_1(0), a(1), c_1(1)* 20 to 21 labelled a__f_1(1), g_1(1), f_1(2)* 20 to 26 labelled c_1(2)* 21 to 15 labelled mark_1(1), a(1), c_1(1)* 21 to 21 labelled a__f_1(1), g_1(1), f_1(2)* 21 to 26 labelled c_1(2)* 22 to 23 labelled f_1(1)* 23 to 24 labelled g_1(1)* 24 to 25 labelled f_1(1)* 25 to 15 labelled a(1)* 26 to 27 labelled f_1(2)* 27 to 28 labelled g_1(2)* 28 to 29 labelled f_1(2)* 29 to 15 labelled a(2)


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(2)
YES