(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(c(b(x))) → b(a(b(a(x))))
b(x) → c(a(c(x)))
a(a(x)) → a(b(c(a(x))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 37, 38, 41, 42, 45, 46, 47, 51, 52, 53, 56, 57, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133
Node 1 is start node and node 3 is final node.
Those nodes are connected through the following edges:
- 1 to 25 labelled b_1(0)
- 1 to 28 labelled c_1(0)
- 1 to 30 labelled a_1(0)
- 1 to 33 labelled c_1(1)
- 3 to 3 labelled #_1(0)
- 25 to 26 labelled a_1(0)
- 25 to 77 labelled b_1(1)
- 25 to 80 labelled c_1(2)
- 26 to 27 labelled b_1(0)
- 26 to 37 labelled c_1(1)
- 27 to 3 labelled a_1(0)
- 27 to 45 labelled b_1(1)
- 27 to 51 labelled a_1(1)
- 27 to 56 labelled c_1(2)
- 28 to 29 labelled a_1(0)
- 28 to 45 labelled b_1(1)
- 28 to 56 labelled c_1(2)
- 29 to 3 labelled c_1(0)
- 30 to 31 labelled b_1(0)
- 30 to 41 labelled c_1(1)
- 31 to 32 labelled c_1(0)
- 32 to 3 labelled a_1(0)
- 32 to 45 labelled b_1(1)
- 32 to 51 labelled a_1(1)
- 32 to 56 labelled c_1(2)
- 33 to 35 labelled a_1(1)
- 33 to 89 labelled b_1(2)
- 33 to 98 labelled c_1(3)
- 35 to 25 labelled c_1(1)
- 37 to 38 labelled a_1(1)
- 37 to 65 labelled b_1(2)
- 37 to 70 labelled c_1(3)
- 38 to 27 labelled c_1(1)
- 41 to 42 labelled a_1(1)
- 42 to 31 labelled c_1(1)
- 45 to 46 labelled a_1(1)
- 45 to 92 labelled b_1(2)
- 45 to 100 labelled c_1(3)
- 46 to 47 labelled b_1(1)
- 46 to 61 labelled c_1(2)
- 47 to 3 labelled a_1(1)
- 47 to 45 labelled b_1(1)
- 47 to 51 labelled a_1(1)
- 47 to 56 labelled c_1(2)
- 51 to 52 labelled b_1(1)
- 51 to 63 labelled c_1(2)
- 52 to 53 labelled c_1(1)
- 53 to 3 labelled a_1(1)
- 53 to 45 labelled b_1(1)
- 53 to 51 labelled a_1(1)
- 53 to 56 labelled c_1(2)
- 56 to 57 labelled a_1(2)
- 56 to 118 labelled b_1(3)
- 56 to 121 labelled c_1(4)
- 57 to 45 labelled c_1(2)
- 61 to 62 labelled a_1(2)
- 61 to 65 labelled b_1(2)
- 61 to 70 labelled c_1(3)
- 62 to 47 labelled c_1(2)
- 63 to 64 labelled a_1(2)
- 64 to 52 labelled c_1(2)
- 65 to 66 labelled a_1(2)
- 66 to 67 labelled b_1(2)
- 66 to 72 labelled c_1(3)
- 67 to 45 labelled a_1(2)
- 67 to 74 labelled a_1(2)
- 70 to 71 labelled a_1(3)
- 71 to 65 labelled c_1(3)
- 72 to 73 labelled a_1(3)
- 73 to 67 labelled c_1(3)
- 74 to 75 labelled b_1(2)
- 74 to 82 labelled c_1(3)
- 75 to 76 labelled c_1(2)
- 76 to 46 labelled a_1(2)
- 76 to 95 labelled b_1(3)
- 76 to 102 labelled c_1(4)
- 77 to 78 labelled a_1(1)
- 78 to 79 labelled b_1(1)
- 78 to 84 labelled c_1(2)
- 79 to 65 labelled a_1(1)
- 79 to 86 labelled a_1(2)
- 80 to 81 labelled a_1(2)
- 81 to 77 labelled c_1(2)
- 82 to 83 labelled a_1(3)
- 83 to 75 labelled c_1(3)
- 84 to 85 labelled a_1(2)
- 85 to 79 labelled c_1(2)
- 86 to 87 labelled b_1(2)
- 86 to 104 labelled c_1(3)
- 87 to 88 labelled c_1(2)
- 88 to 66 labelled a_1(2)
- 89 to 90 labelled a_1(2)
- 90 to 91 labelled b_1(2)
- 90 to 106 labelled c_1(3)
- 91 to 77 labelled a_1(2)
- 91 to 115 labelled a_1(2)
- 92 to 93 labelled a_1(2)
- 93 to 94 labelled b_1(2)
- 93 to 108 labelled c_1(3)
- 94 to 65 labelled a_1(2)
- 94 to 112 labelled a_1(3)
- 95 to 96 labelled a_1(3)
- 96 to 97 labelled b_1(3)
- 96 to 110 labelled c_1(4)
- 97 to 65 labelled a_1(3)
- 97 to 112 labelled a_1(3)
- 98 to 99 labelled a_1(3)
- 99 to 89 labelled c_1(3)
- 100 to 101 labelled a_1(3)
- 101 to 92 labelled c_1(3)
- 102 to 103 labelled a_1(4)
- 103 to 95 labelled c_1(4)
- 104 to 105 labelled a_1(3)
- 105 to 87 labelled c_1(3)
- 106 to 107 labelled a_1(3)
- 107 to 91 labelled c_1(3)
- 108 to 109 labelled a_1(3)
- 109 to 94 labelled c_1(3)
- 110 to 111 labelled a_1(4)
- 111 to 97 labelled c_1(4)
- 112 to 113 labelled b_1(3)
- 112 to 123 labelled c_1(4)
- 113 to 114 labelled c_1(3)
- 114 to 66 labelled a_1(3)
- 115 to 116 labelled b_1(2)
- 115 to 125 labelled c_1(3)
- 116 to 117 labelled c_1(2)
- 117 to 78 labelled a_1(2)
- 118 to 119 labelled a_1(3)
- 119 to 120 labelled b_1(3)
- 119 to 127 labelled c_1(4)
- 120 to 92 labelled a_1(3)
- 120 to 129 labelled a_1(3)
- 121 to 122 labelled a_1(4)
- 122 to 118 labelled c_1(4)
- 123 to 124 labelled a_1(4)
- 124 to 113 labelled c_1(4)
- 125 to 126 labelled a_1(3)
- 126 to 116 labelled c_1(3)
- 127 to 128 labelled a_1(4)
- 128 to 120 labelled c_1(4)
- 129 to 130 labelled b_1(3)
- 129 to 132 labelled c_1(4)
- 130 to 131 labelled c_1(3)
- 131 to 93 labelled a_1(3)
- 132 to 133 labelled a_1(4)
- 133 to 130 labelled c_1(4)