(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:
0(1(2(1(x)))) → 1(2(1(1(0(1(2(0(1(2(x))))))))))
0(1(2(1(x)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(x)))))))))))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 22, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 68, 69, 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, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193
Node 1 is start node and node 2 is final node.
Those nodes are connected through the following edges:
- 1 to 14 labelled 1_1(0)
- 1 to 35 labelled 1_1(0)
- 2 to 2 labelled #_1(0)
- 14 to 15 labelled 2_1(0)
- 15 to 16 labelled 1_1(0)
- 16 to 17 labelled 1_1(0)
- 17 to 18 labelled 0_1(0)
- 17 to 89 labelled 1_1(1)
- 17 to 98 labelled 1_1(1)
- 18 to 19 labelled 1_1(0)
- 19 to 20 labelled 2_1(0)
- 20 to 21 labelled 0_1(0)
- 20 to 68 labelled 1_1(1)
- 20 to 77 labelled 1_1(1)
- 21 to 22 labelled 1_1(0)
- 22 to 2 labelled 2_1(0)
- 35 to 36 labelled 2_1(0)
- 36 to 37 labelled 1_1(0)
- 37 to 38 labelled 1_1(0)
- 38 to 39 labelled 0_1(0)
- 38 to 131 labelled 1_1(1)
- 38 to 140 labelled 1_1(1)
- 39 to 40 labelled 1_1(0)
- 40 to 41 labelled 2_1(0)
- 41 to 42 labelled 0_1(0)
- 41 to 89 labelled 1_1(1)
- 41 to 98 labelled 1_1(1)
- 42 to 43 labelled 1_1(0)
- 43 to 44 labelled 2_1(0)
- 44 to 45 labelled 0_1(0)
- 44 to 68 labelled 1_1(1)
- 44 to 77 labelled 1_1(1)
- 45 to 46 labelled 1_1(0)
- 46 to 2 labelled 2_1(0)
- 68 to 69 labelled 2_1(1)
- 69 to 70 labelled 1_1(1)
- 70 to 71 labelled 1_1(1)
- 71 to 72 labelled 0_1(1)
- 71 to 152 labelled 1_1(2)
- 71 to 161 labelled 1_1(2)
- 72 to 73 labelled 1_1(1)
- 73 to 74 labelled 2_1(1)
- 74 to 75 labelled 0_1(1)
- 74 to 68 labelled 1_1(1)
- 74 to 77 labelled 1_1(1)
- 75 to 76 labelled 1_1(1)
- 76 to 2 labelled 2_1(1)
- 77 to 78 labelled 2_1(1)
- 78 to 79 labelled 1_1(1)
- 79 to 80 labelled 1_1(1)
- 80 to 81 labelled 0_1(1)
- 80 to 173 labelled 1_1(2)
- 80 to 182 labelled 1_1(2)
- 81 to 82 labelled 1_1(1)
- 82 to 83 labelled 2_1(1)
- 83 to 84 labelled 0_1(1)
- 83 to 152 labelled 1_1(2)
- 83 to 161 labelled 1_1(2)
- 84 to 85 labelled 1_1(1)
- 85 to 86 labelled 2_1(1)
- 86 to 87 labelled 0_1(1)
- 86 to 68 labelled 1_1(1)
- 86 to 77 labelled 1_1(1)
- 87 to 88 labelled 1_1(1)
- 88 to 2 labelled 2_1(1)
- 89 to 90 labelled 2_1(1)
- 90 to 91 labelled 1_1(1)
- 91 to 92 labelled 1_1(1)
- 92 to 93 labelled 0_1(1)
- 93 to 94 labelled 1_1(1)
- 94 to 95 labelled 2_1(1)
- 95 to 96 labelled 0_1(1)
- 96 to 97 labelled 1_1(1)
- 97 to 68 labelled 2_1(1)
- 97 to 77 labelled 2_1(1)
- 98 to 99 labelled 2_1(1)
- 99 to 100 labelled 1_1(1)
- 100 to 101 labelled 1_1(1)
- 101 to 102 labelled 0_1(1)
- 102 to 103 labelled 1_1(1)
- 103 to 104 labelled 2_1(1)
- 104 to 105 labelled 0_1(1)
- 105 to 106 labelled 1_1(1)
- 106 to 107 labelled 2_1(1)
- 107 to 108 labelled 0_1(1)
- 108 to 109 labelled 1_1(1)
- 109 to 68 labelled 2_1(1)
- 109 to 77 labelled 2_1(1)
- 131 to 132 labelled 2_1(1)
- 132 to 133 labelled 1_1(1)
- 133 to 134 labelled 1_1(1)
- 134 to 135 labelled 0_1(1)
- 135 to 136 labelled 1_1(1)
- 136 to 137 labelled 2_1(1)
- 137 to 138 labelled 0_1(1)
- 138 to 139 labelled 1_1(1)
- 139 to 89 labelled 2_1(1)
- 139 to 98 labelled 2_1(1)
- 140 to 141 labelled 2_1(1)
- 141 to 142 labelled 1_1(1)
- 142 to 143 labelled 1_1(1)
- 143 to 144 labelled 0_1(1)
- 144 to 145 labelled 1_1(1)
- 145 to 146 labelled 2_1(1)
- 146 to 147 labelled 0_1(1)
- 147 to 148 labelled 1_1(1)
- 148 to 149 labelled 2_1(1)
- 149 to 150 labelled 0_1(1)
- 150 to 151 labelled 1_1(1)
- 151 to 89 labelled 2_1(1)
- 151 to 98 labelled 2_1(1)
- 152 to 153 labelled 2_1(2)
- 153 to 154 labelled 1_1(2)
- 154 to 155 labelled 1_1(2)
- 155 to 156 labelled 0_1(2)
- 156 to 157 labelled 1_1(2)
- 157 to 158 labelled 2_1(2)
- 158 to 159 labelled 0_1(2)
- 159 to 160 labelled 1_1(2)
- 160 to 68 labelled 2_1(2)
- 160 to 77 labelled 2_1(2)
- 161 to 162 labelled 2_1(2)
- 162 to 163 labelled 1_1(2)
- 163 to 164 labelled 1_1(2)
- 164 to 165 labelled 0_1(2)
- 165 to 166 labelled 1_1(2)
- 166 to 167 labelled 2_1(2)
- 167 to 168 labelled 0_1(2)
- 168 to 169 labelled 1_1(2)
- 169 to 170 labelled 2_1(2)
- 170 to 171 labelled 0_1(2)
- 171 to 172 labelled 1_1(2)
- 172 to 68 labelled 2_1(2)
- 172 to 77 labelled 2_1(2)
- 173 to 174 labelled 2_1(2)
- 174 to 175 labelled 1_1(2)
- 175 to 176 labelled 1_1(2)
- 176 to 177 labelled 0_1(2)
- 177 to 178 labelled 1_1(2)
- 178 to 179 labelled 2_1(2)
- 179 to 180 labelled 0_1(2)
- 180 to 181 labelled 1_1(2)
- 181 to 152 labelled 2_1(2)
- 181 to 161 labelled 2_1(2)
- 182 to 183 labelled 2_1(2)
- 183 to 184 labelled 1_1(2)
- 184 to 185 labelled 1_1(2)
- 185 to 186 labelled 0_1(2)
- 186 to 187 labelled 1_1(2)
- 187 to 188 labelled 2_1(2)
- 188 to 189 labelled 0_1(2)
- 189 to 190 labelled 1_1(2)
- 190 to 191 labelled 2_1(2)
- 191 to 192 labelled 0_1(2)
- 192 to 193 labelled 1_1(2)
- 193 to 152 labelled 2_1(2)
- 193 to 161 labelled 2_1(2)