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(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.
↳ QTRS
↳ RFCMatchBoundsTRSProof
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.
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(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:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 2 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)
- 1 to 3 labelled h_1(0), f_1(0), n__h_1(1), n__f_1(1), h_1(1), f_1(1), n__h_1(2), n__f_1(2)
- 1 to 4 labelled g_1(0)
- 1 to 6 labelled g_1(1)
- 1 to 11 labelled g_1(2)
- 1 to 1 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
- 2 to 2 labelled #_1(0)
- 3 to 2 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)
- 3 to 8 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2), g_1(2)
- 3 to 9 labelled g_1(2)
- 3 to 11 labelled g_1(2)
- 3 to 1 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
- 4 to 5 labelled n__h_1(0)
- 5 to 2 labelled n__f_1(0)
- 6 to 7 labelled n__h_1(1)
- 7 to 3 labelled n__f_1(1)
- 7 to 2 labelled n__f_1(1)
- 8 to 2 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)
- 8 to 1 labelled h_1(1), f_1(1), n__h_1(2), n__f_1(2)
- 8 to 8 labelled g_1(2)
- 9 to 10 labelled n__h_1(2)
- 10 to 8 labelled n__f_1(2)
- 11 to 12 labelled n__h_1(2)
- 12 to 2 labelled n__f_1(2)
- 12 to 3 labelled n__f_1(2)
- 12 to 1 labelled n__f_1(2)