Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 RFCMatchBoundsTRSProof (⇔)
↳2 TRUE

(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:

480102, 480103, 480104, 480105, 480106, 480108, 480107, 480109, 480110, 480111, 480113, 480112, 480115, 480114

Node 480102 is start node and node 480103 is final node.

Those nodes are connect through the following edges:

  • 480102 to 480103 labelled g_1(0), n__h_1(0), f_1(0), h_1(0), n__f_1(0), activate_1(0), n__h_1(1), n__f_1(1), g_1(1), f_1(1), h_1(1), activate_1(1), n__f_1(2), n__h_1(2)
  • 480102 to 480104 labelled h_1(0), f_1(0), n__f_1(1), n__h_1(1), h_1(1), f_1(1), n__f_1(2), n__h_1(2)
  • 480102 to 480105 labelled g_1(0)
  • 480102 to 480107 labelled g_1(1)
  • 480102 to 480112 labelled g_1(2)
  • 480102 to 480102 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 480102 to 480114 labelled g_1(2)
  • 480103 to 480103 labelled #_1(0)
  • 480104 to 480103 labelled activate_1(0), g_1(1), f_1(1), h_1(1), n__f_1(1), n__h_1(1), activate_1(1), n__f_1(2), n__h_1(2)
  • 480104 to 480109 labelled h_1(1), f_1(1), n__f_1(2), n__h_1(2)
  • 480104 to 480110 labelled g_1(2)
  • 480104 to 480112 labelled g_1(2)
  • 480104 to 480102 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 480104 to 480114 labelled g_1(2)
  • 480105 to 480106 labelled n__h_1(0)
  • 480106 to 480103 labelled n__f_1(0)
  • 480108 to 480104 labelled n__f_1(1)
  • 480108 to 480103 labelled n__f_1(1)
  • 480107 to 480108 labelled n__h_1(1)
  • 480109 to 480103 labelled activate_1(1), g_1(1), f_1(1), h_1(1), n__f_1(1), n__h_1(1), n__h_1(2), n__f_1(2)
  • 480109 to 480102 labelled h_1(1), f_1(1), n__f_1(2), n__h_1(2)
  • 480109 to 480114 labelled g_1(2)
  • 480109 to 480112 labelled g_1(2)
  • 480110 to 480111 labelled n__h_1(2)
  • 480111 to 480109 labelled n__f_1(2)
  • 480113 to 480103 labelled n__f_1(2)
  • 480113 to 480104 labelled n__f_1(2)
  • 480112 to 480113 labelled n__h_1(2)
  • 480115 to 480102 labelled n__f_1(2)
  • 480114 to 480115 labelled n__h_1(2)

(2) TRUE