(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] 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, 4, 7, 8, 9, 10, 13, 14, 15, 16, 17, 23, 24, 25, 26, 27, 28, 29, 30, 31
Node 1 is start node and node 4 is final node.
Those nodes are connected through the following edges:
- 1 to 7 labelled f_1(0)
- 1 to 9 labelled f_1(0)
- 1 to 4 labelled f_1(0), c_1(0), d_1(0), g_1(0), h_1(0), 0(0), 1(0), f_1(1), c_1(1), d_1(1), g_1(1), h_1(1), 0(1), 1(1), 1(2)
- 1 to 13 labelled g_1(0)
- 1 to 15 labelled g_1(0)
- 1 to 23 labelled f_1(1)
- 1 to 25 labelled f_1(1)
- 1 to 27 labelled g_1(1)
- 1 to 29 labelled g_1(1)
- 1 to 17 labelled h_1(1)
- 1 to 31 labelled h_1(2)
- 4 to 4 labelled #_1(0)
- 7 to 8 labelled c_1(0)
- 8 to 4 labelled f_1(0)
- 8 to 23 labelled f_1(1)
- 8 to 25 labelled f_1(1)
- 9 to 10 labelled d_1(0)
- 10 to 4 labelled f_1(0)
- 10 to 23 labelled f_1(1)
- 10 to 25 labelled f_1(1)
- 13 to 14 labelled d_1(0)
- 14 to 4 labelled 1(0)
- 15 to 16 labelled d_1(0)
- 16 to 17 labelled h_1(0)
- 17 to 4 labelled 0(0)
- 23 to 24 labelled c_1(1)
- 24 to 4 labelled f_1(1)
- 24 to 23 labelled f_1(1)
- 24 to 25 labelled f_1(1)
- 25 to 26 labelled d_1(1)
- 26 to 4 labelled f_1(1)
- 26 to 23 labelled f_1(1)
- 26 to 25 labelled f_1(1)
- 27 to 28 labelled d_1(1)
- 28 to 4 labelled 1(1)
- 29 to 30 labelled d_1(1)
- 30 to 31 labelled h_1(1)
- 31 to 4 labelled 0(1)