YES
proof of Transformed_CSR_04_Ex18_Luc06_FR.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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(f(a)) -> f(g(n__f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__a) -> a
   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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20

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

Those nodes are connected through the following edges:

* 1 to 12 labelled f_1(0), n__f_1(1)* 1 to 2 labelled n__f_1(0), n__a(0), a(0), f_1(0), g_1(0), activate_1(0), n__a(1), n__f_1(1), a(1), f_1(1), g_1(1), activate_1(1), n__a(2), n__f_1(2)* 1 to 15 labelled f_1(1), n__f_1(2)* 1 to 1 labelled f_1(1), n__f_1(2)* 1 to 18 labelled f_1(2), n__f_1(3)* 2 to 2 labelled #_1(0)* 12 to 13 labelled g_1(0)* 12 to 2 labelled activate_1(0), a(1), f_1(1), g_1(1), n__f_1(1), n__a(1), activate_1(1), n__a(2), n__f_1(2)* 12 to 1 labelled f_1(1), n__f_1(2)* 12 to 15 labelled f_1(1), n__f_1(2)* 12 to 18 labelled f_1(2), n__f_1(3)* 13 to 14 labelled n__f_1(0)* 14 to 2 labelled n__a(0)* 15 to 16 labelled g_1(1)* 15 to 2 labelled activate_1(1), a(1), f_1(1), g_1(1), n__f_1(1), n__a(1), n__a(2), n__f_1(2)* 15 to 1 labelled f_1(1), n__f_1(2)* 15 to 15 labelled f_1(1), n__f_1(2)* 15 to 18 labelled f_1(2), n__f_1(3)* 16 to 17 labelled n__f_1(1)* 17 to 2 labelled n__a(1)* 18 to 19 labelled g_1(2)* 19 to 20 labelled n__f_1(2)* 20 to 2 labelled n__a(2)


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(2)
YES