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:

8857833, 8857834, 8857835, 8857837, 8857836, 8857839, 8857838, 8857840, 8857841, 8857842

Node 8857833 is start node and node 8857834 is final node.

Those nodes are connect through the following edges:

  • 8857833 to 8857834 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__h_1(2), n__f_1(2)
  • 8857833 to 8857835 labelled h_1(0), f_1(0), n__h_1(1), n__f_1(1)
  • 8857833 to 8857836 labelled g_1(0)
  • 8857833 to 8857838 labelled g_1(1)
  • 8857833 to 8857840 labelled h_1(1), f_1(1), n__f_1(2), n__h_1(2)
  • 8857833 to 8857841 labelled g_1(2)
  • 8857833 to 8857833 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 8857834 to 8857834 labelled #_1(0)
  • 8857835 to 8857834 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__h_1(2), n__f_1(2)
  • 8857835 to 8857833 labelled h_1(1), f_1(1), n__f_1(2), n__h_1(2)
  • 8857835 to 8857841 labelled g_1(2)
  • 8857837 to 8857834 labelled n__f_1(0)
  • 8857836 to 8857837 labelled n__h_1(0)
  • 8857839 to 8857834 labelled n__f_1(1)
  • 8857839 to 8857835 labelled n__f_1(1)
  • 8857838 to 8857839 labelled n__h_1(1)
  • 8857840 to 8857834 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)
  • 8857840 to 8857833 labelled h_1(1), f_1(1), n__f_1(2), n__h_1(2)
  • 8857840 to 8857841 labelled g_1(2)
  • 8857841 to 8857842 labelled n__h_1(2)
  • 8857842 to 8857833 labelled n__f_1(2)
  • 8857842 to 8857834 labelled n__f_1(2)
  • 8857842 to 8857840 labelled n__f_1(2)

(2) TRUE