(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. This implies Q-termination of R.
The following rules were used to construct the certificate:
active(f(f(a))) → mark(f(g(f(a))))
mark(f(X)) → active(f(mark(X)))
mark(a) → active(a)
mark(g(X)) → active(g(X))
f(mark(X)) → f(X)
f(active(X)) → f(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, 4, 5, 8, 10, 12, 13, 14, 19, 20, 21, 22, 28, 29, 30, 31, 32, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51
Node 2 is start node and node 4 is final node.
Those nodes are connected through the following edges:
- 2 to 5 labelled mark_1(0)
- 2 to 13 labelled active_1(0)
- 2 to 12 labelled active_1(0)
- 2 to 4 labelled f_1(0), g_1(0), f_1(1), g_1(1)
- 2 to 19 labelled active_1(1)
- 2 to 29 labelled mark_1(1)
- 2 to 39 labelled active_1(2)
- 4 to 4 labelled #_1(0)
- 5 to 8 labelled f_1(0)
- 8 to 10 labelled g_1(0)
- 10 to 12 labelled f_1(0)
- 12 to 4 labelled a(0)
- 13 to 14 labelled f_1(0)
- 13 to 4 labelled g_1(0), g_1(1), f_1(1)
- 13 to 21 labelled f_1(1)
- 13 to 13 labelled f_1(1)
- 13 to 29 labelled f_1(1)
- 13 to 39 labelled f_1(1)
- 13 to 41 labelled f_1(1)
- 13 to 46 labelled f_1(1)
- 14 to 4 labelled mark_1(0)
- 14 to 21 labelled active_1(1)
- 14 to 13 labelled active_1(1)
- 14 to 29 labelled mark_1(1)
- 14 to 39 labelled active_1(2)
- 14 to 41 labelled mark_1(2)
- 14 to 46 labelled active_1(3)
- 19 to 20 labelled f_1(1)
- 19 to 8 labelled f_1(2)
- 19 to 28 labelled f_1(2)
- 20 to 8 labelled mark_1(1)
- 20 to 28 labelled active_1(1)
- 21 to 22 labelled f_1(1)
- 21 to 4 labelled a(1), f_1(2), f_1(1)
- 21 to 21 labelled f_1(2)
- 21 to 13 labelled f_1(2)
- 21 to 29 labelled f_1(2)
- 21 to 41 labelled f_1(2)
- 21 to 39 labelled f_1(2)
- 21 to 46 labelled f_1(2)
- 22 to 4 labelled mark_1(1)
- 22 to 21 labelled active_1(1)
- 22 to 13 labelled active_1(1)
- 22 to 29 labelled mark_1(1)
- 22 to 41 labelled mark_1(2)
- 22 to 39 labelled active_1(2)
- 22 to 46 labelled active_1(3)
- 28 to 10 labelled g_1(1)
- 29 to 30 labelled f_1(1)
- 30 to 31 labelled g_1(1)
- 31 to 32 labelled f_1(1)
- 32 to 4 labelled a(1)
- 39 to 40 labelled f_1(2)
- 39 to 30 labelled f_1(3)
- 39 to 48 labelled f_1(3)
- 40 to 30 labelled mark_1(2)
- 40 to 48 labelled active_1(2)
- 41 to 42 labelled f_1(2)
- 42 to 43 labelled g_1(2)
- 43 to 44 labelled f_1(2)
- 44 to 4 labelled a(2)
- 46 to 47 labelled f_1(3)
- 46 to 42 labelled f_1(4)
- 46 to 51 labelled f_1(4)
- 47 to 42 labelled mark_1(3)
- 47 to 51 labelled active_1(3)
- 48 to 31 labelled g_1(2)
- 51 to 43 labelled g_1(3)