YES
proof of Rubio_04_gmnp.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(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(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 10, 12, 21, 22, 23, 24, 25, 26

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

Those nodes are connected through the following edges:

* 1 to 3 labelled f_1(0)* 1 to 2 labelled f_1(0), a(0), c_1(0), d_1(0), b(0), e_1(0), g_1(0), f_1(1), a(1), c_1(1), d_1(1), b(1), e_1(1), g_1(1), a(2), b(2), b(3)* 1 to 10 labelled f_1(0)* 1 to 21 labelled f_1(1)* 1 to 23 labelled f_1(1)* 1 to 25 labelled f_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled c_1(0)* 4 to 2 labelled a(0)* 10 to 12 labelled d_1(0)* 12 to 2 labelled b(0), a(0)* 21 to 22 labelled c_1(1)* 22 to 2 labelled a(1)* 23 to 24 labelled d_1(1)* 24 to 2 labelled b(1), a(1)* 25 to 26 labelled d_1(2)* 26 to 2 labelled b(2)


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(2)
YES