YES
proof of Rubio_04_nestrec.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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


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

   f(g(X)) -> g(f(f(X)))
   f(h(X)) -> h(g(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:

   g(f(X)) -> f(f(g(X)))
   h(f(X)) -> g(h(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 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
63, 64, 65, 66, 67, 80, 81, 82

Node 63 is start node and node 64 is final node.

Those nodes are connected through the following edges:

* 63 to 65 labelled f_1(0)* 63 to 67 labelled g_1(0)* 64 to 64 labelled #_1(0)* 65 to 66 labelled f_1(0)* 66 to 64 labelled g_1(0)* 66 to 80 labelled f_1(1)* 67 to 64 labelled h_1(0)* 67 to 82 labelled g_1(1)* 80 to 81 labelled f_1(1)* 81 to 64 labelled g_1(1)* 81 to 80 labelled f_1(1)* 82 to 64 labelled h_1(1)* 82 to 82 labelled g_1(1)


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(4)
YES