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:

2333084, 2333085, 2333086, 2333088, 2333087, 2333089, 2333090, 2333091, 2333092, 2333093

Node 2333084 is start node and node 2333085 is final node.

Those nodes are connect through the following edges:

  • 2333084 to 2333085 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)
  • 2333084 to 2333086 labelled h_1(0), f_1(0), n__f_1(1), n__h_1(1), h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 2333084 to 2333087 labelled g_1(0)
  • 2333084 to 2333089 labelled g_1(1)
  • 2333084 to 2333092 labelled g_1(2)
  • 2333084 to 2333084 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 2333085 to 2333085 labelled #_1(0)
  • 2333086 to 2333085 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)
  • 2333086 to 2333091 labelled h_1(1), f_1(1), n__f_1(2), n__h_1(2)
  • 2333086 to 2333092 labelled g_1(2)
  • 2333086 to 2333084 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 2333088 to 2333085 labelled n__f_1(0)
  • 2333087 to 2333088 labelled n__h_1(0)
  • 2333089 to 2333090 labelled n__h_1(1)
  • 2333090 to 2333086 labelled n__f_1(1)
  • 2333090 to 2333085 labelled n__f_1(1)
  • 2333091 to 2333085 labelled activate_1(1), g_1(1), f_1(1), h_1(1), n__f_1(1), n__h_1(1), n__f_1(2), n__h_1(2)
  • 2333091 to 2333084 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
  • 2333091 to 2333092 labelled g_1(2)
  • 2333092 to 2333093 labelled n__h_1(2)
  • 2333093 to 2333085 labelled n__f_1(2)
  • 2333093 to 2333091 labelled n__f_1(2)
  • 2333093 to 2333086 labelled n__f_1(2)
  • 2333093 to 2333084 labelled n__f_1(2)

(2) TRUE