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:
g(c(x1)) → g(f(c(x1)))
g(f(c(x1))) → g(f(f(c(x1))))
g(g(x1)) → g(f(g(x1)))
f(f(g(x1))) → g(f(x1))
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
g(c(x1)) → g(f(c(x1)))
g(f(c(x1))) → g(f(f(c(x1))))
g(g(x1)) → g(f(g(x1)))
f(f(g(x1))) → g(f(x1))
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:
g(c(x1)) → g(f(c(x1)))
g(f(c(x1))) → g(f(f(c(x1))))
g(g(x1)) → g(f(g(x1)))
f(f(g(x1))) → g(f(x1))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 6, 4, 5, 8, 7, 11, 9, 10, 12, 14, 13, 17, 15, 16, 19, 18, 21, 20
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled g_1(0)
- 1 to 4 labelled g_1(0)
- 1 to 5 labelled g_1(0)
- 1 to 9 labelled g_1(1)
- 1 to 12 labelled g_1(1)
- 1 to 13 labelled g_1(2)
- 1 to 18 labelled g_1(2)
- 1 to 7 labelled g_1(1)
- 2 to 2 labelled #_1(0), g_1(0)
- 2 to 7 labelled g_1(1)
- 2 to 9 labelled g_1(1)
- 2 to 10 labelled g_1(1)
- 2 to 12 labelled g_1(1)
- 2 to 15 labelled g_1(2)
- 2 to 18 labelled g_1(2)
- 2 to 13 labelled g_1(2)
- 3 to 2 labelled f_1(0)
- 3 to 12 labelled g_1(1)
- 3 to 9 labelled g_1(1)
- 3 to 18 labelled g_1(2)
- 3 to 13 labelled g_1(2)
- 6 to 2 labelled c_1(0)
- 4 to 5 labelled f_1(0)
- 5 to 6 labelled f_1(0)
- 8 to 2 labelled g_1(1)
- 8 to 7 labelled g_1(1)
- 8 to 9 labelled g_1(1)
- 8 to 10 labelled g_1(1)
- 8 to 13 labelled g_1(2), g_1(1)
- 8 to 15 labelled g_1(1), g_1(2)
- 8 to 18 labelled g_1(2)
- 7 to 8 labelled f_1(1)
- 11 to 2 labelled c_1(1)
- 9 to 10 labelled f_1(1)
- 10 to 11 labelled f_1(1)
- 10 to 12 labelled g_1(1)
- 10 to 18 labelled g_1(2), g_1(1)
- 10 to 9 labelled g_1(1)
- 10 to 13 labelled g_1(2)
- 12 to 2 labelled f_1(1)
- 12 to 12 labelled g_1(1)
- 12 to 18 labelled g_1(2)
- 12 to 9 labelled g_1(1)
- 12 to 13 labelled g_1(2)
- 14 to 7 labelled g_1(2)
- 14 to 9 labelled g_1(2)
- 14 to 10 labelled g_1(2)
- 14 to 15 labelled g_1(2)
- 14 to 18 labelled g_1(2)
- 14 to 20 labelled g_1(3)
- 14 to 13 labelled g_1(2)
- 13 to 14 labelled f_1(2)
- 17 to 2 labelled c_1(2)
- 15 to 16 labelled f_1(2)
- 16 to 17 labelled f_1(2)
- 19 to 12 labelled g_1(2)
- 19 to 20 labelled g_1(3)
- 19 to 13 labelled g_1(2)
- 19 to 18 labelled g_1(2)
- 19 to 9 labelled g_1(1)
- 18 to 19 labelled f_1(2)
- 21 to 18 labelled g_1(3)
- 21 to 13 labelled g_1(3)
- 20 to 21 labelled f_1(3)