YES
proof of Transformed_CSR_04_Ex25_Luc06_Z.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 34 ms]
(4) YES


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

   f(f(X)) -> c(n__f(g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> n__f(g(n__f(c(X))))
   c(X) -> activate(d(X))
   h(X) -> n__d(c(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   n__f(activate(X)) -> f(X)
   n__d(activate(X)) -> d(X)
   activate(X) -> X

Q is empty.

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(3) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R.
The following rules were used to construct the certificate:

   f(f(X)) -> n__f(g(n__f(c(X))))
   c(X) -> activate(d(X))
   h(X) -> n__d(c(X))
   f(X) -> n__f(X)
   d(X) -> n__d(X)
   n__f(activate(X)) -> f(X)
   n__d(activate(X)) -> d(X)
   activate(X) -> X

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

Node 15 is start node and node 16 is final node.

Those nodes are connected through the following edges:

* 15 to 17 labelled n__f_1(0)* 15 to 20 labelled activate_1(0)* 15 to 19 labelled n__d_1(0)* 15 to 16 labelled n__f_1(0), n__d_1(0), f_1(0), d_1(0), g_1(0), c_1(0), activate_1(0), h_1(0), n__f_1(1), n__d_1(1), d_1(1), f_1(1), g_1(1), c_1(1), activate_1(1), h_1(1), n__d_1(2), n__f_1(2), c_1(2), d_1(2), n__d_1(3), d_1(3), n__d_1(4), d_1(4), n__d_1(5)* 15 to 21 labelled activate_1(1), d_1(1), n__d_1(2)* 15 to 22 labelled n__f_1(1)* 15 to 25 labelled activate_1(2), f_1(1), d_1(1), activate_1(1), f_1(2), d_1(2), d_1(3), f_1(3), n__f_1(2), n__d_1(2), n__f_1(3), n__d_1(3), n__d_1(4), n__f_1(4), n__d_1(1), d_1(4), n__d_1(5)* 15 to 26 labelled activate_1(3), f_1(1), d_1(1), activate_1(1), f_1(2), d_1(2), activate_1(2), d_1(3), f_1(3), d_1(4), n__f_1(2), n__d_1(2), n__f_1(3), n__d_1(3), n__d_1(4), n__f_1(4), n__d_1(5), n__d_1(1)* 16 to 16 labelled #_1(0), c_1(1), c_1(2), d_1(3), n__d_1(4), n__d_1(3), d_1(4), n__d_1(5), d_1(1), d_1(2), n__d_1(2)* 16 to 25 labelled activate_1(2), d_1(3), n__d_1(4), n__d_1(3), d_1(4), n__d_1(5)* 16 to 26 labelled activate_1(3), d_1(4), d_1(3), n__d_1(5), n__d_1(4), n__d_1(3)* 17 to 18 labelled g_1(0)* 18 to 19 labelled n__f_1(0)* 18 to 21 labelled f_1(1), n__f_1(2)* 19 to 16 labelled c_1(0), d_1(2), n__d_1(3), n__d_1(2), d_1(1)* 19 to 21 labelled activate_1(1)* 19 to 25 labelled d_1(3), d_1(2), n__d_1(4), n__d_1(3), n__d_1(2)* 19 to 26 labelled d_1(4), d_1(3), n__d_1(5), n__d_1(4), d_1(2), n__d_1(3), n__d_1(2)* 20 to 16 labelled d_1(0), n__d_1(1), d_1(1), n__d_1(2)* 20 to 25 labelled d_1(2), n__d_1(3), d_1(3), n__d_1(4)* 20 to 26 labelled d_1(2), d_1(3), n__d_1(3), n__d_1(4)* 21 to 16 labelled d_1(1), n__d_1(2)* 21 to 25 labelled d_1(3), n__d_1(4)* 21 to 26 labelled d_1(3), n__d_1(4)* 22 to 23 labelled g_1(1)* 23 to 24 labelled n__f_1(1)* 23 to 25 labelled f_1(2), n__f_1(3)* 24 to 16 labelled c_1(1), d_1(3), n__d_1(4), n__d_1(3), d_1(1), d_1(2), n__d_1(2)* 24 to 25 labelled activate_1(2), d_1(3), n__d_1(4), n__d_1(3)* 24 to 26 labelled d_1(4), d_1(3), n__d_1(5), n__d_1(4), n__d_1(3)* 25 to 16 labelled d_1(2), n__d_1(3), d_1(1), n__d_1(2)* 25 to 25 labelled d_1(3), n__d_1(4)* 25 to 26 labelled d_1(4), n__d_1(5), d_1(3), n__d_1(4)* 26 to 16 labelled d_1(3), n__d_1(4), d_1(1), n__d_1(2)* 26 to 25 labelled d_1(3), n__d_1(4)* 26 to 26 labelled d_1(4), n__d_1(5), d_1(3), n__d_1(4)


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(4)
YES