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(n__f(n__a)) → f(n__g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(activate(X))
activate(X) → X
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(n__f(n__a)) → f(n__g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(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(n__f(n__a)) → f(n__g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(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
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__g_1(0), f_1(0), n__f_1(0), n__a(0), a(0), activate_1(0), n__g_1(1), n__a(1), n__f_1(1), g_1(1), f_1(1), a(1), activate_1(1), n__g_1(2), n__a(2), n__f_1(2)
- 1 to 3 labelled f_1(0), n__f_1(1)
- 1 to 1 labelled g_1(0), n__g_1(1), g_1(1), n__g_1(2)
- 1 to 6 labelled f_1(1), n__f_1(2)
- 2 to 2 labelled #_1(0)
- 3 to 4 labelled n__g_1(0)
- 4 to 5 labelled n__f_1(0)
- 5 to 2 labelled n__a(0)
- 6 to 7 labelled n__g_1(1)
- 7 to 8 labelled n__f_1(1)
- 8 to 2 labelled n__a(1)