(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(b(a(a(b(x))))) → c(c(c(c(a(x)))))
c(x) → b(a(a(b(x))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
3, 4, 12, 13, 14, 15, 16, 17, 18, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 87, 89, 90, 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, 134, 135, 136, 137, 141, 142, 143, 144, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318
Node 3 is start node and node 4 is final node.
Those nodes are connected through the following edges:
- 3 to 12 labelled c_1(0)
- 3 to 16 labelled b_1(0)
- 3 to 29 labelled b_1(1)
- 4 to 4 labelled #_1(0)
- 12 to 13 labelled c_1(0)
- 12 to 32 labelled b_1(1)
- 13 to 14 labelled c_1(0)
- 13 to 35 labelled b_1(1)
- 14 to 15 labelled c_1(0)
- 14 to 38 labelled b_1(1)
- 15 to 4 labelled a_1(0)
- 15 to 41 labelled c_1(1)
- 15 to 45 labelled b_1(2)
- 16 to 17 labelled a_1(0)
- 16 to 75 labelled c_1(1)
- 16 to 79 labelled b_1(2)
- 17 to 18 labelled a_1(0)
- 17 to 41 labelled c_1(1)
- 17 to 45 labelled b_1(2)
- 18 to 4 labelled b_1(0)
- 29 to 30 labelled a_1(1)
- 30 to 31 labelled a_1(1)
- 31 to 12 labelled b_1(1)
- 32 to 33 labelled a_1(1)
- 33 to 34 labelled a_1(1)
- 34 to 13 labelled b_1(1)
- 35 to 36 labelled a_1(1)
- 36 to 37 labelled a_1(1)
- 37 to 14 labelled b_1(1)
- 38 to 39 labelled a_1(1)
- 38 to 95 labelled c_1(2)
- 38 to 103 labelled b_1(3)
- 39 to 40 labelled a_1(1)
- 39 to 41 labelled c_1(1)
- 39 to 45 labelled b_1(2)
- 40 to 15 labelled b_1(1)
- 41 to 42 labelled c_1(1)
- 41 to 48 labelled b_1(2)
- 42 to 43 labelled c_1(1)
- 42 to 51 labelled b_1(2)
- 43 to 44 labelled c_1(1)
- 43 to 56 labelled b_1(2)
- 44 to 4 labelled a_1(1)
- 44 to 41 labelled c_1(1)
- 44 to 45 labelled b_1(2)
- 45 to 46 labelled a_1(2)
- 46 to 47 labelled a_1(2)
- 47 to 41 labelled b_1(2)
- 48 to 49 labelled a_1(2)
- 49 to 50 labelled a_1(2)
- 50 to 42 labelled b_1(2)
- 51 to 52 labelled a_1(2)
- 52 to 53 labelled a_1(2)
- 53 to 43 labelled b_1(2)
- 56 to 57 labelled a_1(2)
- 56 to 127 labelled c_1(3)
- 56 to 175 labelled b_1(4)
- 57 to 58 labelled a_1(2)
- 57 to 41 labelled c_1(1)
- 57 to 45 labelled b_1(2)
- 58 to 44 labelled b_1(2)
- 75 to 76 labelled c_1(1)
- 75 to 82 labelled b_1(2)
- 76 to 77 labelled c_1(1)
- 76 to 87 labelled b_1(2)
- 77 to 78 labelled c_1(1)
- 77 to 92 labelled b_1(2)
- 78 to 41 labelled a_1(1)
- 78 to 99 labelled c_1(2)
- 78 to 106 labelled b_1(3)
- 79 to 80 labelled a_1(2)
- 80 to 81 labelled a_1(2)
- 81 to 75 labelled b_1(2)
- 82 to 83 labelled a_1(2)
- 83 to 84 labelled a_1(2)
- 84 to 76 labelled b_1(2)
- 87 to 89 labelled a_1(2)
- 89 to 90 labelled a_1(2)
- 90 to 77 labelled b_1(2)
- 92 to 93 labelled a_1(2)
- 93 to 94 labelled a_1(2)
- 94 to 78 labelled b_1(2)
- 95 to 96 labelled c_1(2)
- 95 to 109 labelled b_1(3)
- 96 to 97 labelled c_1(2)
- 96 to 112 labelled b_1(3)
- 97 to 98 labelled c_1(2)
- 97 to 115 labelled b_1(3)
- 98 to 41 labelled a_1(2)
- 98 to 134 labelled c_1(3)
- 98 to 178 labelled b_1(4)
- 99 to 100 labelled c_1(2)
- 99 to 118 labelled b_1(3)
- 100 to 101 labelled c_1(2)
- 100 to 121 labelled b_1(3)
- 101 to 102 labelled c_1(2)
- 101 to 124 labelled b_1(3)
- 102 to 42 labelled a_1(2)
- 102 to 141 labelled c_1(3)
- 102 to 181 labelled b_1(4)
- 103 to 104 labelled a_1(3)
- 104 to 105 labelled a_1(3)
- 105 to 95 labelled b_1(3)
- 106 to 107 labelled a_1(3)
- 107 to 108 labelled a_1(3)
- 108 to 99 labelled b_1(3)
- 109 to 110 labelled a_1(3)
- 110 to 111 labelled a_1(3)
- 111 to 96 labelled b_1(3)
- 112 to 113 labelled a_1(3)
- 113 to 114 labelled a_1(3)
- 114 to 97 labelled b_1(3)
- 115 to 116 labelled a_1(3)
- 116 to 117 labelled a_1(3)
- 117 to 98 labelled b_1(3)
- 118 to 119 labelled a_1(3)
- 119 to 120 labelled a_1(3)
- 120 to 100 labelled b_1(3)
- 121 to 122 labelled a_1(3)
- 122 to 123 labelled a_1(3)
- 123 to 101 labelled b_1(3)
- 124 to 125 labelled a_1(3)
- 125 to 126 labelled a_1(3)
- 126 to 102 labelled b_1(3)
- 127 to 128 labelled c_1(3)
- 127 to 184 labelled b_1(4)
- 128 to 129 labelled c_1(3)
- 128 to 187 labelled b_1(4)
- 129 to 130 labelled c_1(3)
- 129 to 282 labelled b_1(4)
- 130 to 41 labelled a_1(3)
- 130 to 134 labelled c_1(3)
- 130 to 178 labelled b_1(4)
- 134 to 135 labelled c_1(3)
- 134 to 285 labelled b_1(4)
- 135 to 136 labelled c_1(3)
- 135 to 288 labelled b_1(4)
- 136 to 137 labelled c_1(3)
- 136 to 291 labelled b_1(4)
- 137 to 42 labelled a_1(3)
- 137 to 141 labelled c_1(3)
- 137 to 181 labelled b_1(4)
- 141 to 142 labelled c_1(3)
- 141 to 294 labelled b_1(4)
- 142 to 143 labelled c_1(3)
- 142 to 297 labelled b_1(4)
- 143 to 144 labelled c_1(3)
- 143 to 300 labelled b_1(4)
- 144 to 43 labelled a_1(3)
- 144 to 303 labelled c_1(3)
- 144 to 307 labelled b_1(4)
- 175 to 176 labelled a_1(4)
- 176 to 177 labelled a_1(4)
- 177 to 127 labelled b_1(4)
- 178 to 179 labelled a_1(4)
- 179 to 180 labelled a_1(4)
- 180 to 134 labelled b_1(4)
- 181 to 182 labelled a_1(4)
- 182 to 183 labelled a_1(4)
- 183 to 141 labelled b_1(4)
- 184 to 185 labelled a_1(4)
- 185 to 186 labelled a_1(4)
- 186 to 128 labelled b_1(4)
- 187 to 188 labelled a_1(4)
- 188 to 189 labelled a_1(4)
- 189 to 129 labelled b_1(4)
- 282 to 283 labelled a_1(4)
- 283 to 284 labelled a_1(4)
- 284 to 130 labelled b_1(4)
- 285 to 286 labelled a_1(4)
- 286 to 287 labelled a_1(4)
- 287 to 135 labelled b_1(4)
- 288 to 289 labelled a_1(4)
- 289 to 290 labelled a_1(4)
- 290 to 136 labelled b_1(4)
- 291 to 292 labelled a_1(4)
- 292 to 293 labelled a_1(4)
- 293 to 137 labelled b_1(4)
- 294 to 295 labelled a_1(4)
- 295 to 296 labelled a_1(4)
- 296 to 142 labelled b_1(4)
- 297 to 298 labelled a_1(4)
- 298 to 299 labelled a_1(4)
- 299 to 143 labelled b_1(4)
- 300 to 301 labelled a_1(4)
- 301 to 302 labelled a_1(4)
- 302 to 144 labelled b_1(4)
- 303 to 304 labelled c_1(3)
- 303 to 310 labelled b_1(4)
- 304 to 305 labelled c_1(3)
- 304 to 313 labelled b_1(4)
- 305 to 306 labelled c_1(3)
- 305 to 316 labelled b_1(4)
- 306 to 44 labelled a_1(3)
- 306 to 127 labelled c_1(3)
- 306 to 175 labelled b_1(4)
- 307 to 308 labelled a_1(4)
- 308 to 309 labelled a_1(4)
- 309 to 303 labelled b_1(4)
- 310 to 311 labelled a_1(4)
- 311 to 312 labelled a_1(4)
- 312 to 304 labelled b_1(4)
- 313 to 314 labelled a_1(4)
- 314 to 315 labelled a_1(4)
- 315 to 305 labelled b_1(4)
- 316 to 317 labelled a_1(4)
- 316 to 127 labelled c_1(3)
- 316 to 175 labelled b_1(4)
- 317 to 318 labelled a_1(4)
- 317 to 41 labelled c_1(1)
- 317 to 45 labelled b_1(2)
- 318 to 306 labelled b_1(4)