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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   f(n__f(n__a)) -> f(n__g(f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   g(X) -> n__g(X)
   activate(n__f(X)) -> f(X)
   activate(n__a) -> a
   activate(n__g(X)) -> g(X)
   activate(X) -> 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:

   f(n__f(n__a)) -> f(n__g(f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   g(X) -> n__g(X)
   activate(n__f(X)) -> f(X)
   activate(n__a) -> a
   activate(n__g(X)) -> g(X)
   activate(X) -> X

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 11, 12, 13, 14, 15, 16

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

Those nodes are connected through the following edges:

* 1 to 11 labelled f_1(0), n__f_1(1)* 1 to 2 labelled n__f_1(0), n__a(0), n__g_1(0), f_1(0), a(0), g_1(0), activate_1(0), n__f_1(1), n__a(1), n__g_1(1), f_1(1), a(1), g_1(1), activate_1(1), n__f_1(2), n__a(2), n__g_1(2)* 1 to 14 labelled f_1(1), n__f_1(2)* 2 to 2 labelled #_1(0)* 11 to 12 labelled n__g_1(0)* 12 to 13 labelled f_1(0), n__f_1(1)* 13 to 2 labelled n__a(0)* 14 to 15 labelled n__g_1(1)* 15 to 16 labelled f_1(1), n__f_1(2)* 16 to 2 labelled n__a(1)


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(2)
YES