YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex18_Luc06_Z.ari

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f(f(a)) → f(g(n__f(a)))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X

Q is empty.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

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(a)) → f(g(n__f(a)))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 3, 6, 8, 10, 14, 15, 16

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

Those nodes are connected through the following edges:

  • 1 to 6 labelled f_1(0), n__f_1(1)
  • 1 to 3 labelled n__f_1(0), f_1(0), a(0), g_1(0), activate_1(0), n__f_1(1), f_1(1), a(1), g_1(1), activate_1(1), n__f_1(2)
  • 1 to 14 labelled f_1(1), n__f_1(2)
  • 3 to 3 labelled #_1(0)
  • 6 to 8 labelled g_1(0)
  • 8 to 10 labelled n__f_1(0)
  • 10 to 3 labelled a(0)
  • 14 to 15 labelled g_1(1)
  • 15 to 16 labelled n__f_1(1)
  • 16 to 3 labelled a(1)

(2) YES