(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(a(x)) → a(b(b(b(x))))
b(a(x)) → b(b(c(x)))
a(b(b(c(x)))) → a(a(a(b(x))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 72, 73, 74, 75, 76, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97
Node 1 is start node and node 2 is final node.
Those nodes are connected through the following edges:
- 1 to 5 labelled a_1(0)
- 1 to 8 labelled b_1(0)
- 1 to 10 labelled a_1(0)
- 1 to 45 labelled a_1(1)
- 2 to 2 labelled #_1(0)
- 5 to 6 labelled b_1(0)
- 6 to 7 labelled b_1(0)
- 7 to 2 labelled b_1(0)
- 7 to 48 labelled b_1(1)
- 8 to 9 labelled b_1(0)
- 9 to 2 labelled c_1(0)
- 10 to 11 labelled a_1(0)
- 10 to 50 labelled a_1(1)
- 11 to 12 labelled a_1(0)
- 11 to 53 labelled a_1(1)
- 11 to 58 labelled a_1(2)
- 12 to 2 labelled b_1(0)
- 12 to 48 labelled b_1(1)
- 45 to 46 labelled b_1(1)
- 46 to 47 labelled b_1(1)
- 47 to 11 labelled b_1(1)
- 47 to 50 labelled b_1(1)
- 47 to 56 labelled b_1(2)
- 48 to 49 labelled b_1(1)
- 49 to 2 labelled c_1(1)
- 50 to 51 labelled b_1(1)
- 51 to 52 labelled b_1(1)
- 51 to 12 labelled c_1(1)
- 52 to 12 labelled b_1(1)
- 52 to 53 labelled b_1(1)
- 52 to 58 labelled b_1(1)
- 52 to 72 labelled b_1(2)
- 53 to 54 labelled a_1(1)
- 53 to 61 labelled a_1(2)
- 54 to 55 labelled a_1(1)
- 54 to 53 labelled a_1(1)
- 54 to 58 labelled a_1(2)
- 54 to 74 labelled a_1(2)
- 54 to 88 labelled a_1(3)
- 55 to 2 labelled b_1(1)
- 55 to 48 labelled b_1(1)
- 56 to 57 labelled b_1(2)
- 57 to 53 labelled c_1(2)
- 57 to 58 labelled c_1(2)
- 58 to 59 labelled b_1(2)
- 59 to 60 labelled b_1(2)
- 60 to 54 labelled b_1(2)
- 60 to 61 labelled b_1(2)
- 60 to 56 labelled b_1(2)
- 60 to 86 labelled b_1(3)
- 61 to 62 labelled b_1(2)
- 62 to 63 labelled b_1(2)
- 62 to 55 labelled c_1(2)
- 63 to 55 labelled b_1(2)
- 63 to 53 labelled b_1(2)
- 63 to 72 labelled b_1(2)
- 63 to 84 labelled b_1(3)
- 63 to 58 labelled b_1(2)
- 63 to 74 labelled b_1(2)
- 63 to 88 labelled b_1(2)
- 72 to 73 labelled b_1(2)
- 73 to 54 labelled c_1(2)
- 73 to 61 labelled c_1(2)
- 74 to 75 labelled a_1(2)
- 74 to 91 labelled a_1(3)
- 74 to 61 labelled a_1(2)
- 75 to 76 labelled a_1(2)
- 75 to 53 labelled a_1(1)
- 75 to 58 labelled a_1(2)
- 75 to 74 labelled a_1(2)
- 75 to 88 labelled a_1(3)
- 76 to 2 labelled b_1(2)
- 76 to 48 labelled b_1(1)
- 84 to 85 labelled b_1(3)
- 85 to 61 labelled c_1(3)
- 85 to 75 labelled c_1(3)
- 85 to 91 labelled c_1(3)
- 86 to 87 labelled b_1(3)
- 87 to 58 labelled c_1(3)
- 87 to 74 labelled c_1(3)
- 87 to 88 labelled c_1(3)
- 88 to 89 labelled b_1(3)
- 89 to 90 labelled b_1(3)
- 90 to 75 labelled b_1(3)
- 90 to 91 labelled b_1(3)
- 90 to 56 labelled b_1(2)
- 90 to 61 labelled b_1(3)
- 90 to 86 labelled b_1(3)
- 90 to 96 labelled b_1(4)
- 91 to 92 labelled b_1(3)
- 92 to 93 labelled b_1(3)
- 92 to 76 labelled c_1(3)
- 93 to 76 labelled b_1(3)
- 93 to 58 labelled b_1(3)
- 93 to 74 labelled b_1(3)
- 93 to 84 labelled b_1(3)
- 93 to 94 labelled b_1(4)
- 93 to 88 labelled b_1(3)
- 94 to 95 labelled b_1(4)
- 95 to 91 labelled c_1(4)
- 96 to 97 labelled b_1(4)
- 97 to 88 labelled c_1(4)