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(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)
Q is empty.
Termination of the TRS R could be shown with a Match Bound [6,7] of 3. This implies Q-termination of R.
The following rules were used to construct the certificate:
f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(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), a(0), c_1(0), e_1(0), b(0), f_1(0), d_1(0), e_1(1), g_1(1), a(1), c_1(1), b(1), f_1(1), d_1(1), b(2), a(2), b(3)
- 1 to 3 labelled f_1(0)
- 1 to 5 labelled f_1(0)
- 1 to 7 labelled f_1(1)
- 1 to 9 labelled f_1(1)
- 1 to 11 labelled f_1(2)
- 2 to 2 labelled #_1(0)
- 3 to 4 labelled c_1(0)
- 4 to 2 labelled a(0)
- 5 to 6 labelled d_1(0)
- 6 to 2 labelled b(0), a(0)
- 7 to 8 labelled c_1(1)
- 8 to 2 labelled a(1)
- 9 to 10 labelled d_1(1)
- 10 to 2 labelled b(1), a(1)
- 11 to 12 labelled d_1(2)
- 12 to 2 labelled b(2)