(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(a)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 3, 10, 11, 12, 13, 14, 16, 17, 18, 19, 24
Node 1 is start node and node 3 is final node.
Those nodes are connected through the following edges:
- 1 to 10 labelled c_1(0)
- 1 to 3 labelled n__f_1(0), n__g_1(0), n__a(0), a(0), f_1(0), c_1(0), g_1(0), activate_1(0), n__a(1), n__f_1(1), n__g_1(1), a(1), f_1(1), c_1(1), g_1(1), activate_1(1), n__a(2), n__f_1(2), n__g_1(2)
- 1 to 14 labelled f_1(0), g_1(0), n__f_1(1), n__g_1(1), c_1(2)
- 1 to 16 labelled c_1(1)
- 1 to 24 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)
- 1 to 1 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)
- 3 to 3 labelled #_1(0)
- 10 to 11 labelled n__f_1(0)
- 11 to 12 labelled n__g_1(0)
- 12 to 13 labelled n__f_1(0)
- 13 to 3 labelled n__a(0)
- 14 to 3 labelled activate_1(0), a(1), f_1(1), c_1(1), n__f_1(1), n__g_1(1), n__a(1), g_1(1), activate_1(1), n__a(2), n__f_1(2), n__g_1(2)
- 14 to 1 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)
- 14 to 16 labelled c_1(1)
- 14 to 14 labelled c_1(2)
- 16 to 17 labelled n__f_1(1)
- 17 to 18 labelled n__g_1(1)
- 18 to 19 labelled n__f_1(1)
- 19 to 3 labelled n__a(1)
- 24 to 3 labelled activate_1(1), a(1), f_1(1), c_1(1), n__f_1(1), n__g_1(1), n__a(1), g_1(1), n__a(2), n__f_1(2), n__g_1(2)
- 24 to 1 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)
- 24 to 16 labelled c_1(1)
- 24 to 14 labelled c_1(2)