YES Termination w.r.t. Q proof of Der95_06.ari

(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.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

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, 3, 5, 6, 9, 10

Node 1 is start node and node 3 is final node.

Those nodes are connected through the following edges:

  • 1 to 5 labelled g_1(0)
  • 3 to 3 labelled #_1(0)
  • 5 to 6 labelled g_1(0)
  • 6 to 3 labelled f_1(0), g_1(0)
  • 6 to 9 labelled g_1(1)
  • 9 to 10 labelled g_1(1)
  • 10 to 3 labelled f_1(1), g_1(1)
  • 10 to 9 labelled g_1(1)

(2) YES