YES
proof of Rubio_04_aoto.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 0 ms]
(4) YES


----------------------------------------

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

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))

Q is empty.

----------------------------------------

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> g(f(X))

Q is empty.

----------------------------------------

(3) 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(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> g(f(X))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
3, 4, 5, 6, 7, 8, 13, 14, 15, 17, 21, 23, 24

Node 3 is start node and node 4 is final node.

Those nodes are connected through the following edges:

* 3 to 5 labelled f_1(0)* 3 to 8 labelled g_1(0)* 3 to 21 labelled g_1(1)* 4 to 4 labelled #_1(0)* 5 to 6 labelled g_1(0)* 6 to 7 labelled f_1(0)* 6 to 17 labelled g_1(1)* 6 to 8 labelled g_1(1)* 7 to 8 labelled g_1(0)* 8 to 4 labelled f_1(0)* 8 to 13 labelled f_1(1)* 8 to 17 labelled g_1(1)* 8 to 23 labelled g_1(2)* 13 to 14 labelled g_1(1)* 14 to 15 labelled f_1(1)* 14 to 24 labelled g_1(2)* 15 to 17 labelled g_1(1)* 17 to 4 labelled f_1(1)* 17 to 13 labelled f_1(1)* 17 to 17 labelled g_1(1)* 17 to 23 labelled g_1(2)* 21 to 7 labelled f_1(1)* 21 to 17 labelled g_1(1)* 21 to 8 labelled g_1(1)* 23 to 15 labelled f_1(2)* 23 to 24 labelled g_1(2)* 24 to 4 labelled f_1(2)* 24 to 13 labelled f_1(1), f_1(2)* 24 to 17 labelled g_1(1)* 24 to 23 labelled g_1(2)


----------------------------------------

(4)
YES