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(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0))) → g(d(1))
g(c(1)) → g(d(h(0)))
g(h(x)) → g(x)
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0))) → g(d(1))
g(c(1)) → g(d(h(0)))
g(h(x)) → 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(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0))) → g(d(1))
g(c(1)) → g(d(h(0)))
g(h(x)) → g(x)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 5, 3, 4, 6, 7, 8, 9, 12, 10, 11, 13, 14, 15, 16
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 2 labelled f_1(0), h_1(0), 1(0), c_1(0), g_1(0), 0(0), d_1(0), f_1(1), h_1(1), 1(1), c_1(1), g_1(1), 0(1), d_1(1), 1(2)
- 1 to 3 labelled g_1(0)
- 1 to 6 labelled f_1(0)
- 1 to 8 labelled f_1(0)
- 1 to 10 labelled g_1(1)
- 1 to 13 labelled f_1(1)
- 1 to 15 labelled f_1(1)
- 1 to 5 labelled h_1(1)
- 1 to 12 labelled h_1(2)
- 2 to 2 labelled #_1(0)
- 5 to 2 labelled 0(0)
- 3 to 4 labelled d_1(0)
- 4 to 5 labelled h_1(0)
- 4 to 2 labelled 1(0)
- 6 to 7 labelled c_1(0)
- 7 to 2 labelled f_1(0)
- 7 to 13 labelled f_1(1)
- 7 to 15 labelled f_1(1)
- 8 to 9 labelled d_1(0)
- 9 to 2 labelled f_1(0)
- 9 to 13 labelled f_1(1)
- 9 to 15 labelled f_1(1)
- 12 to 2 labelled 0(1)
- 10 to 11 labelled d_1(1)
- 11 to 12 labelled h_1(1)
- 11 to 2 labelled 1(1)
- 13 to 14 labelled c_1(1)
- 14 to 2 labelled f_1(1)
- 14 to 13 labelled f_1(1)
- 14 to 15 labelled f_1(1)
- 15 to 16 labelled d_1(1)
- 16 to 2 labelled f_1(1)
- 16 to 13 labelled f_1(1)
- 16 to 15 labelled f_1(1)