YES Termination w.r.t. Q proof of Transformed_CSR_04_ExConc_Zan97_FR.ari

(0) Obligation:

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

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(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(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 4, 5, 8, 9, 13, 14, 16, 19, 20

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

Those nodes are connected through the following edges:

  • 1 to 5 labelled g_1(0)
  • 1 to 4 labelled n__h_1(0), n__f_1(0), f_1(0), g_1(0), h_1(0), activate_1(0), n__f_1(1), n__h_1(1), f_1(1), g_1(1), h_1(1), activate_1(1), n__f_1(2), n__h_1(2)
  • 1 to 9 labelled h_1(0), f_1(0), n__h_1(1), n__f_1(1)
  • 1 to 13 labelled g_1(1)
  • 1 to 16 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 1 to 19 labelled g_1(2)
  • 1 to 1 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 4 to 4 labelled #_1(0)
  • 5 to 8 labelled n__h_1(0)
  • 8 to 4 labelled n__f_1(0)
  • 9 to 4 labelled activate_1(0), f_1(1), g_1(1), n__h_1(1), n__f_1(1), h_1(1), activate_1(1), n__f_1(2), n__h_1(2)
  • 9 to 1 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 9 to 19 labelled g_1(2)
  • 13 to 14 labelled n__h_1(1)
  • 14 to 4 labelled n__f_1(1)
  • 14 to 9 labelled n__f_1(1)
  • 16 to 4 labelled activate_1(1), f_1(1), g_1(1), n__h_1(1), n__f_1(1), h_1(1), n__f_1(2), n__h_1(2)
  • 16 to 1 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 16 to 9 labelled g_1(2)
  • 16 to 19 labelled g_1(2)
  • 19 to 20 labelled n__h_1(2)
  • 20 to 4 labelled n__f_1(2)
  • 20 to 16 labelled n__f_1(2)
  • 20 to 1 labelled n__f_1(2)

(2) YES