YES
proof of /TPDB/Der95_06.ari
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


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

   f(g(x)) -> g(g(f(x)))
   f(g(x)) -> g(g(g(x)))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 4, 6, 8, 10, 11

Node 1 is start node and node 4 is final node.

Those nodes are connected through the following edges:

* 1 to 6 labelled g_1(0)* 4 to 4 labelled #_1(0)* 6 to 8 labelled g_1(0)* 8 to 4 labelled f_1(0), g_1(0)* 8 to 10 labelled g_1(1)* 10 to 11 labelled g_1(1)* 11 to 4 labelled f_1(1), g_1(1)* 11 to 10 labelled g_1(1)


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(2)
YES