(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 6. This implies Q-termination of R.
The following rules were used to construct the certificate:
a'(f(f(active(x)))) → a'(f(g(f(mark(x)))))
f(active(X)) → active(f(X))
mark(f(X)) → f(mark(X))
f(proper(X)) → proper(f(X))
a'(proper(x)) → a'(ok(x))
g(proper(X)) → proper(g(X))
ok(f(X)) → f(ok(X))
ok(g(X)) → g(ok(X))
mark(top(X)) → proper(top(X))
ok(top(X)) → active(top(X))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 3, 5, 10, 11, 12, 14, 20, 22, 23, 24, 28, 31, 33, 34, 35, 36, 37, 39, 40, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 61, 62, 63, 66, 68, 72, 73, 74, 75, 79, 80
Node 1 is start node and node 3 is final node.
Those nodes are connected through the following edges:
- 1 to 5 labelled a'_1(0), f_1(0), g_1(0)
- 1 to 14 labelled active_1(0), proper_1(0)
- 1 to 12 labelled f_1(0)
- 1 to 28 labelled active_1(1), proper_1(1)
- 1 to 34 labelled a'_1(1)
- 1 to 57 labelled a'_1(2)
- 3 to 3 labelled #_1(0)
- 5 to 10 labelled f_1(0)
- 5 to 3 labelled ok_1(0)
- 5 to 20 labelled f_1(1), g_1(1)
- 5 to 22 labelled active_1(1)
- 5 to 31 labelled active_1(2)
- 5 to 39 labelled proper_1(1)
- 10 to 11 labelled g_1(0)
- 10 to 33 labelled proper_1(1)
- 11 to 12 labelled f_1(0)
- 11 to 28 labelled proper_1(1)
- 12 to 3 labelled mark_1(0)
- 12 to 23 labelled f_1(1)
- 12 to 22 labelled proper_1(1)
- 12 to 31 labelled proper_1(2)
- 14 to 3 labelled f_1(0), g_1(0), top_1(0)
- 14 to 24 labelled active_1(1), proper_1(1)
- 20 to 3 labelled ok_1(1)
- 20 to 20 labelled f_1(1), g_1(1)
- 20 to 22 labelled active_1(1)
- 20 to 31 labelled active_1(2)
- 22 to 3 labelled top_1(1)
- 23 to 3 labelled mark_1(1)
- 23 to 23 labelled f_1(1)
- 23 to 22 labelled proper_1(1)
- 23 to 31 labelled proper_1(2)
- 24 to 3 labelled f_1(1), g_1(1)
- 24 to 24 labelled active_1(1), proper_1(1)
- 28 to 22 labelled f_1(1)
- 28 to 31 labelled f_1(1)
- 28 to 39 labelled f_1(1), g_1(1)
- 31 to 22 labelled f_1(2)
- 31 to 31 labelled f_1(2)
- 33 to 28 labelled g_1(1)
- 34 to 35 labelled f_1(1)
- 34 to 39 labelled ok_1(1)
- 34 to 50 labelled f_1(2)
- 34 to 53 labelled proper_1(2)
- 35 to 36 labelled g_1(1)
- 35 to 51 labelled proper_1(2)
- 36 to 37 labelled f_1(1)
- 36 to 47 labelled proper_1(2)
- 37 to 22 labelled mark_1(1)
- 37 to 40 labelled proper_1(2)
- 37 to 31 labelled mark_1(1)
- 37 to 48 labelled f_1(2)
- 37 to 54 labelled proper_1(3)
- 39 to 33 labelled f_1(1)
- 40 to 3 labelled top_1(2)
- 47 to 40 labelled f_1(2)
- 47 to 54 labelled f_1(2)
- 48 to 22 labelled mark_1(2)
- 48 to 31 labelled mark_1(2)
- 48 to 40 labelled proper_1(2)
- 48 to 52 labelled f_1(3)
- 48 to 54 labelled proper_1(3)
- 48 to 62 labelled proper_1(4)
- 50 to 33 labelled ok_1(2)
- 50 to 56 labelled g_1(2)
- 51 to 47 labelled g_1(2)
- 52 to 22 labelled mark_1(3)
- 52 to 31 labelled mark_1(3)
- 52 to 40 labelled proper_1(2)
- 52 to 52 labelled f_1(3)
- 52 to 54 labelled proper_1(3)
- 52 to 62 labelled proper_1(4)
- 53 to 51 labelled f_1(2)
- 54 to 40 labelled f_1(3)
- 54 to 54 labelled f_1(3)
- 54 to 62 labelled f_1(3)
- 56 to 28 labelled ok_1(2)
- 56 to 58 labelled f_1(2), g_1(2)
- 56 to 54 labelled active_1(3)
- 57 to 53 labelled ok_1(2)
- 57 to 63 labelled f_1(3)
- 58 to 22 labelled ok_1(2)
- 58 to 31 labelled ok_1(2)
- 58 to 39 labelled ok_1(2)
- 58 to 50 labelled f_1(2)
- 58 to 40 labelled active_1(2)
- 58 to 61 labelled f_1(3)
- 58 to 54 labelled active_1(3)
- 58 to 62 labelled active_1(4)
- 61 to 22 labelled ok_1(3)
- 61 to 31 labelled ok_1(3)
- 61 to 40 labelled active_1(2)
- 61 to 61 labelled f_1(3)
- 61 to 54 labelled active_1(3)
- 61 to 62 labelled active_1(4)
- 62 to 54 labelled f_1(4)
- 62 to 62 labelled f_1(4)
- 63 to 51 labelled ok_1(3)
- 63 to 66 labelled g_1(3)
- 66 to 47 labelled ok_1(3)
- 66 to 68 labelled f_1(3)
- 66 to 74 labelled active_1(4)
- 68 to 40 labelled ok_1(3)
- 68 to 54 labelled ok_1(3)
- 68 to 72 labelled active_1(3)
- 68 to 73 labelled f_1(4)
- 68 to 74 labelled active_1(4)
- 68 to 79 labelled active_1(5)
- 72 to 3 labelled top_1(3)
- 73 to 40 labelled ok_1(4)
- 73 to 54 labelled ok_1(4)
- 73 to 62 labelled ok_1(4)
- 73 to 72 labelled active_1(3)
- 73 to 73 labelled f_1(4)
- 73 to 75 labelled f_1(5)
- 73 to 74 labelled active_1(4)
- 73 to 79 labelled active_1(5)
- 73 to 80 labelled active_1(6)
- 74 to 72 labelled f_1(4)
- 74 to 74 labelled f_1(4)
- 74 to 79 labelled f_1(4)
- 75 to 54 labelled ok_1(5)
- 75 to 62 labelled ok_1(5)
- 75 to 73 labelled f_1(4)
- 75 to 75 labelled f_1(5)
- 75 to 74 labelled active_1(4)
- 75 to 79 labelled active_1(5)
- 75 to 80 labelled active_1(6)
- 79 to 74 labelled f_1(5)
- 79 to 79 labelled f_1(5)
- 79 to 80 labelled f_1(5)
- 80 to 79 labelled f_1(6)
- 80 to 80 labelled f_1(6)