Termination w.r.t. Q of the following Term Rewriting System could be proven:
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.
↳ QTRS
↳ RFCMatchBoundsTRSProof
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.
Termination of the TRS R could be shown with a Match Bound [6,7] 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:
1, 2, 3, 6, 7, 4, 5, 8, 11, 12, 9, 10, 13, 14
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled f_1(0)
- 1 to 4 labelled f_1(0)
- 1 to 8 labelled f_1(1)
- 2 to 2 labelled #_1(0)
- 3 to 2 labelled g_1(0)
- 6 to 7 labelled g_1(0)
- 7 to 2 labelled f_1(0)
- 7 to 8 labelled f_1(1)
- 7 to 9 labelled f_1(1)
- 7 to 14 labelled f_1(2)
- 7 to 13 labelled f_1(2)
- 4 to 5 labelled g_1(0)
- 5 to 6 labelled f_1(0)
- 5 to 8 labelled f_1(1)
- 8 to 2 labelled g_1(1)
- 8 to 6 labelled g_1(1)
- 8 to 9 labelled g_1(1)
- 8 to 8 labelled g_1(1)
- 8 to 14 labelled g_1(1)
- 8 to 13 labelled g_1(1)
- 11 to 12 labelled g_1(1)
- 12 to 2 labelled f_1(1)
- 12 to 8 labelled f_1(1)
- 12 to 9 labelled f_1(1)
- 12 to 14 labelled f_1(2)
- 12 to 13 labelled f_1(2)
- 9 to 10 labelled g_1(1)
- 10 to 11 labelled f_1(1)
- 10 to 13 labelled f_1(2)
- 10 to 8 labelled f_1(1)
- 10 to 14 labelled f_1(2)
- 13 to 2 labelled g_1(2)
- 13 to 9 labelled g_1(2)
- 13 to 8 labelled g_1(2)
- 13 to 14 labelled g_1(2)
- 14 to 11 labelled g_1(2)
- 14 to 13 labelled g_1(2)