(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. This implies Q-termination of R.
The following rules were used to construct the certificate:
g(c(x)) → g(f(c(x)))
g(f(c(x))) → g(f(f(c(x))))
g(g(x)) → g(f(g(x)))
f(f(g(x))) → g(f(x))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
8, 9, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 38, 39
Node 8 is start node and node 9 is final node.
Those nodes are connected through the following edges:
- 8 to 10 labelled g_1(0)
- 8 to 12 labelled g_1(0)
- 8 to 24 labelled g_1(1)
- 8 to 22 labelled g_1(1)
- 8 to 33 labelled g_1(2)
- 9 to 9 labelled #_1(0)
- 10 to 11 labelled f_1(0)
- 10 to 9 labelled f_1(0)
- 10 to 21 labelled g_1(1)
- 10 to 24 labelled g_1(1)
- 10 to 36 labelled g_1(2)
- 11 to 9 labelled c_1(0), g_1(0)
- 11 to 22 labelled g_1(1)
- 11 to 24 labelled g_1(1)
- 11 to 33 labelled g_1(2)
- 12 to 13 labelled f_1(0)
- 13 to 14 labelled f_1(0)
- 14 to 9 labelled c_1(0)
- 21 to 9 labelled f_1(1)
- 21 to 21 labelled g_1(1)
- 21 to 36 labelled g_1(2)
- 21 to 24 labelled g_1(1)
- 22 to 23 labelled f_1(1)
- 23 to 9 labelled c_1(1), g_1(1)
- 23 to 22 labelled g_1(1)
- 23 to 24 labelled g_1(1)
- 23 to 33 labelled g_1(2)
- 24 to 25 labelled f_1(1)
- 25 to 26 labelled f_1(1)
- 25 to 21 labelled g_1(1)
- 25 to 36 labelled g_1(2), g_1(1)
- 25 to 24 labelled g_1(1)
- 26 to 9 labelled c_1(1)
- 33 to 34 labelled f_1(2)
- 34 to 35 labelled f_1(2)
- 35 to 9 labelled c_1(2)
- 36 to 37 labelled f_1(2)
- 37 to 21 labelled g_1(2)
- 37 to 36 labelled g_1(2)
- 37 to 38 labelled g_1(3)
- 37 to 24 labelled g_1(2), g_1(1)
- 38 to 39 labelled f_1(3)
- 39 to 36 labelled g_1(3)