(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:
a(d(x)) → d(b(c(b(d(x)))))
a(x) → b(b(f(b(b(x)))))
b(d(b(x))) → a(d(x))
d(f(x)) → b(d(x))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 24, 25, 26, 27, 28, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 148, 149, 150, 151, 152, 153, 154, 155
Node 3 is start node and node 4 is final node.
Those nodes are connected through the following edges:
- 3 to 5 labelled d_1(0)
- 3 to 9 labelled b_1(0)
- 3 to 8 labelled a_1(0), b_1(0)
- 3 to 21 labelled b_1(1)
- 3 to 25 labelled d_1(1)
- 3 to 28 labelled a_1(1)
- 3 to 46 labelled b_1(2)
- 3 to 50 labelled d_1(2)
- 3 to 54 labelled a_1(1)
- 3 to 55 labelled b_1(2)
- 3 to 59 labelled d_1(2)
- 4 to 4 labelled #_1(0)
- 5 to 6 labelled b_1(0)
- 6 to 7 labelled c_1(0)
- 7 to 8 labelled b_1(0)
- 7 to 28 labelled a_1(1)
- 7 to 46 labelled b_1(2)
- 7 to 50 labelled d_1(2)
- 7 to 54 labelled a_1(1)
- 7 to 55 labelled b_1(2)
- 7 to 59 labelled d_1(2)
- 8 to 4 labelled d_1(0)
- 8 to 28 labelled b_1(1), a_1(1)
- 8 to 46 labelled b_1(2)
- 8 to 50 labelled d_1(2)
- 8 to 62 labelled a_1(2)
- 8 to 63 labelled b_1(3)
- 8 to 67 labelled d_1(3)
- 8 to 59 labelled d_1(2)
- 9 to 10 labelled b_1(0)
- 10 to 11 labelled f_1(0)
- 11 to 12 labelled b_1(0)
- 11 to 54 labelled a_1(1)
- 11 to 55 labelled b_1(2)
- 11 to 59 labelled d_1(2)
- 11 to 28 labelled a_1(1)
- 11 to 46 labelled b_1(2)
- 11 to 50 labelled d_1(2)
- 12 to 4 labelled b_1(0)
- 12 to 28 labelled a_1(1)
- 12 to 46 labelled b_1(2)
- 12 to 50 labelled d_1(2)
- 12 to 59 labelled d_1(2)
- 21 to 22 labelled b_1(1)
- 22 to 23 labelled f_1(1)
- 23 to 24 labelled b_1(1)
- 23 to 62 labelled a_1(2)
- 23 to 63 labelled b_1(3)
- 23 to 67 labelled d_1(3)
- 24 to 8 labelled b_1(1)
- 24 to 28 labelled a_1(1)
- 24 to 46 labelled b_1(2)
- 24 to 50 labelled d_1(2)
- 24 to 62 labelled a_1(2)
- 24 to 63 labelled b_1(3)
- 24 to 67 labelled d_1(3)
- 24 to 59 labelled d_1(2)
- 25 to 26 labelled b_1(1)
- 26 to 27 labelled c_1(1)
- 27 to 28 labelled b_1(1), a_1(1)
- 27 to 46 labelled b_1(2)
- 27 to 50 labelled d_1(2)
- 27 to 62 labelled a_1(2)
- 27 to 63 labelled b_1(3)
- 27 to 67 labelled d_1(3)
- 27 to 59 labelled d_1(2)
- 28 to 4 labelled d_1(1)
- 28 to 28 labelled b_1(1), a_1(1)
- 28 to 46 labelled b_1(2)
- 28 to 50 labelled d_1(1), d_1(2)
- 28 to 62 labelled a_1(2)
- 28 to 63 labelled b_1(3)
- 28 to 67 labelled d_1(3), d_1(1)
- 28 to 68 labelled d_1(1)
- 28 to 59 labelled d_1(2), d_1(1)
- 28 to 60 labelled d_1(1)
- 46 to 47 labelled b_1(2)
- 47 to 48 labelled f_1(2)
- 48 to 49 labelled b_1(2)
- 48 to 70 labelled a_1(3)
- 48 to 148 labelled b_1(4)
- 48 to 152 labelled d_1(4)
- 49 to 28 labelled b_1(2), a_1(1)
- 49 to 46 labelled b_1(2)
- 49 to 50 labelled d_1(2)
- 49 to 62 labelled a_1(2)
- 49 to 70 labelled a_1(3)
- 49 to 63 labelled b_1(3)
- 49 to 148 labelled b_1(4)
- 49 to 67 labelled d_1(3)
- 49 to 152 labelled d_1(4)
- 49 to 59 labelled d_1(2)
- 50 to 51 labelled b_1(2)
- 51 to 52 labelled c_1(2)
- 52 to 53 labelled b_1(2)
- 52 to 28 labelled a_1(1)
- 52 to 46 labelled b_1(2)
- 52 to 50 labelled d_1(2)
- 52 to 70 labelled a_1(3)
- 52 to 148 labelled b_1(4)
- 52 to 152 labelled d_1(4)
- 52 to 59 labelled d_1(2)
- 53 to 4 labelled d_1(2)
- 53 to 28 labelled b_1(1), a_1(1)
- 53 to 46 labelled b_1(2)
- 53 to 50 labelled d_1(2)
- 53 to 62 labelled a_1(2)
- 53 to 63 labelled b_1(3)
- 53 to 67 labelled d_1(3), d_1(2)
- 53 to 59 labelled d_1(2)
- 54 to 51 labelled d_1(1)
- 55 to 56 labelled b_1(2)
- 56 to 57 labelled f_1(2)
- 57 to 58 labelled b_1(2)
- 58 to 54 labelled b_1(2)
- 59 to 60 labelled b_1(2)
- 60 to 61 labelled c_1(2)
- 61 to 62 labelled b_1(2)
- 62 to 51 labelled d_1(2)
- 62 to 68 labelled d_1(2)
- 62 to 60 labelled d_1(2)
- 63 to 64 labelled b_1(3)
- 64 to 65 labelled f_1(3)
- 65 to 66 labelled b_1(3)
- 66 to 62 labelled b_1(3)
- 67 to 68 labelled b_1(3)
- 68 to 69 labelled c_1(3)
- 69 to 70 labelled b_1(3)
- 70 to 51 labelled d_1(3)
- 70 to 68 labelled d_1(3)
- 70 to 153 labelled d_1(3)
- 70 to 60 labelled d_1(3)
- 148 to 149 labelled b_1(4)
- 149 to 150 labelled f_1(4)
- 150 to 151 labelled b_1(4)
- 151 to 70 labelled b_1(4)
- 152 to 153 labelled b_1(4)
- 153 to 154 labelled c_1(4)
- 154 to 155 labelled b_1(4)
- 155 to 51 labelled d_1(4)
- 155 to 68 labelled d_1(4)
- 155 to 153 labelled d_1(4)
- 155 to 60 labelled d_1(4)