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(f(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(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(X)) → f(g(f(g(f(X)))))
f(g(f(X))) → f(g(X))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9629143, 9629144, 9629145, 9629149, 9629147, 9629148, 9629146, 9629150, 9629151, 9629152, 9629153, 9629154, 9629155, 9629156, 9629157

Node 9629143 is start node and node 9629144 is final node.

Those nodes are connect through the following edges:

  • 9629143 to 9629145 labelled f_1(0)
  • 9629143 to 9629146 labelled f_1(0)
  • 9629143 to 9629150 labelled f_1(1)
  • 9629144 to 9629144 labelled #_1(0)
  • 9629145 to 9629144 labelled g_1(0)
  • 9629149 to 9629144 labelled f_1(0)
  • 9629149 to 9629150 labelled f_1(1)
  • 9629149 to 9629151 labelled f_1(1)
  • 9629149 to 9629155 labelled f_1(2)
  • 9629149 to 9629157 labelled f_1(2)
  • 9629149 to 9629156 labelled f_1(2)
  • 9629147 to 9629148 labelled f_1(0)
  • 9629147 to 9629150 labelled f_1(1)
  • 9629148 to 9629149 labelled g_1(0)
  • 9629146 to 9629147 labelled g_1(0)
  • 9629150 to 9629144 labelled g_1(1)
  • 9629150 to 9629148 labelled g_1(1)
  • 9629150 to 9629151 labelled g_1(1)
  • 9629150 to 9629150 labelled g_1(1)
  • 9629150 to 9629155 labelled g_1(1)
  • 9629150 to 9629157 labelled g_1(1)
  • 9629150 to 9629156 labelled g_1(1)
  • 9629151 to 9629152 labelled g_1(1)
  • 9629152 to 9629153 labelled f_1(1)
  • 9629152 to 9629156 labelled f_1(2)
  • 9629152 to 9629150 labelled f_1(1)
  • 9629152 to 9629157 labelled f_1(2)
  • 9629153 to 9629154 labelled g_1(1)
  • 9629154 to 9629144 labelled f_1(1)
  • 9629154 to 9629150 labelled f_1(1)
  • 9629154 to 9629151 labelled f_1(1)
  • 9629154 to 9629157 labelled f_1(2)
  • 9629154 to 9629155 labelled f_1(2)
  • 9629154 to 9629156 labelled f_1(2)
  • 9629155 to 9629153 labelled g_1(2)
  • 9629156 to 9629144 labelled g_1(2)
  • 9629156 to 9629151 labelled g_1(2)
  • 9629156 to 9629150 labelled g_1(2)
  • 9629156 to 9629157 labelled g_1(2)
  • 9629156 to 9629155 labelled g_1(2)
  • 9629157 to 9629156 labelled g_1(2)

(2) TRUE