(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:
active(f(f(a))) → mark(c(f(g(f(a)))))
mark(f(X)) → active(f(mark(X)))
mark(a) → active(a)
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
2, 3, 5, 8, 10, 12, 14, 16, 18, 21, 22, 23, 24, 25, 26, 27, 37, 38, 39, 40, 41, 51, 52, 53, 54, 55, 56, 57
Node 2 is start node and node 3 is final node.
Those nodes are connected through the following edges:
- 2 to 5 labelled mark_1(0)
- 2 to 16 labelled active_1(0)
- 2 to 14 labelled active_1(0)
- 2 to 21 labelled active_1(0)
- 2 to 3 labelled f_1(0), c_1(0), g_1(0), f_1(1), c_1(1), g_1(1)
- 2 to 23 labelled active_1(1)
- 2 to 37 labelled mark_1(1)
- 2 to 51 labelled active_1(2)
- 3 to 3 labelled #_1(0)
- 5 to 8 labelled c_1(0)
- 8 to 10 labelled f_1(0)
- 10 to 12 labelled g_1(0)
- 12 to 14 labelled f_1(0)
- 14 to 3 labelled a(0)
- 16 to 18 labelled f_1(0)
- 16 to 3 labelled c_1(0), c_1(1), f_1(1)
- 16 to 24 labelled f_1(1)
- 16 to 16 labelled f_1(1)
- 16 to 26 labelled f_1(1)
- 16 to 37 labelled f_1(1)
- 16 to 51 labelled f_1(1)
- 16 to 52 labelled f_1(1)
- 16 to 57 labelled f_1(1)
- 18 to 3 labelled mark_1(0)
- 18 to 24 labelled active_1(1)
- 18 to 16 labelled active_1(1)
- 18 to 26 labelled active_1(1)
- 18 to 37 labelled mark_1(1)
- 18 to 51 labelled active_1(2)
- 18 to 52 labelled mark_1(2)
- 18 to 57 labelled active_1(3)
- 21 to 22 labelled g_1(0)
- 21 to 3 labelled g_1(1)
- 21 to 24 labelled g_1(1)
- 21 to 16 labelled g_1(1)
- 21 to 26 labelled g_1(1)
- 21 to 37 labelled g_1(1)
- 21 to 51 labelled g_1(1)
- 21 to 52 labelled g_1(1)
- 21 to 57 labelled g_1(1)
- 22 to 3 labelled mark_1(0)
- 22 to 24 labelled active_1(1)
- 22 to 16 labelled active_1(1)
- 22 to 26 labelled active_1(1)
- 22 to 37 labelled mark_1(1)
- 22 to 51 labelled active_1(2)
- 22 to 52 labelled mark_1(2)
- 22 to 57 labelled active_1(3)
- 23 to 8 labelled c_1(1)
- 24 to 25 labelled f_1(1)
- 24 to 3 labelled a(1), f_1(2), f_1(1)
- 24 to 24 labelled f_1(2)
- 24 to 16 labelled f_1(2)
- 24 to 26 labelled f_1(2)
- 24 to 37 labelled f_1(2)
- 24 to 52 labelled f_1(2)
- 24 to 51 labelled f_1(2)
- 24 to 57 labelled f_1(2)
- 25 to 3 labelled mark_1(1)
- 25 to 24 labelled active_1(1)
- 25 to 16 labelled active_1(1)
- 25 to 26 labelled active_1(1)
- 25 to 37 labelled mark_1(1)
- 25 to 52 labelled mark_1(2)
- 25 to 51 labelled active_1(2)
- 25 to 57 labelled active_1(3)
- 26 to 27 labelled g_1(1)
- 26 to 3 labelled g_1(2), g_1(1)
- 26 to 24 labelled g_1(2)
- 26 to 16 labelled g_1(2)
- 26 to 26 labelled g_1(2)
- 26 to 37 labelled g_1(2)
- 26 to 52 labelled g_1(2)
- 26 to 51 labelled g_1(2)
- 26 to 57 labelled g_1(2)
- 27 to 3 labelled mark_1(1)
- 27 to 24 labelled active_1(1)
- 27 to 16 labelled active_1(1)
- 27 to 26 labelled active_1(1)
- 27 to 37 labelled mark_1(1)
- 27 to 52 labelled mark_1(2)
- 27 to 51 labelled active_1(2)
- 27 to 57 labelled active_1(3)
- 37 to 38 labelled c_1(1)
- 38 to 39 labelled f_1(1)
- 39 to 40 labelled g_1(1)
- 40 to 41 labelled f_1(1)
- 41 to 3 labelled a(1)
- 51 to 38 labelled c_1(2)
- 52 to 53 labelled c_1(2)
- 53 to 54 labelled f_1(2)
- 54 to 55 labelled g_1(2)
- 55 to 56 labelled f_1(2)
- 56 to 3 labelled a(2)
- 57 to 53 labelled c_1(3)