YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RFCMatchBoundsTRSProof [EQUIVALENT, 16 ms] (2) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(b(a(a(b(x1))))) -> c(c(c(c(a(x1))))) c(x1) -> b(a(a(b(x1)))) Q is empty. ---------------------------------------- (1) RFCMatchBoundsTRSProof (EQUIVALENT) 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(x1))))) -> c(c(c(c(a(x1))))) c(x1) -> b(a(a(b(x1)))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 1, 2, 5, 7, 8, 10, 13, 15, 18, 19, 20, 22, 25, 27, 29, 31, 33, 34, 37, 38, 40, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 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, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270 Node 1 is start node and node 2 is final node. Those nodes are connected through the following edges: * 1 to 5 labelled c_1(0)* 1 to 13 labelled b_1(0)* 1 to 19 labelled b_1(1)* 2 to 2 labelled #_1(0)* 5 to 7 labelled c_1(0)* 5 to 25 labelled b_1(1)* 7 to 8 labelled c_1(0)* 7 to 31 labelled b_1(1)* 8 to 10 labelled c_1(0)* 8 to 37 labelled b_1(1)* 10 to 2 labelled a_1(0)* 10 to 43 labelled c_1(1)* 10 to 51 labelled b_1(2)* 13 to 15 labelled a_1(0)* 13 to 159 labelled c_1(1)* 13 to 163 labelled b_1(2)* 15 to 18 labelled a_1(0)* 15 to 43 labelled c_1(1)* 15 to 51 labelled b_1(2)* 18 to 2 labelled b_1(0)* 19 to 20 labelled a_1(1)* 20 to 22 labelled a_1(1)* 22 to 5 labelled b_1(1)* 25 to 27 labelled a_1(1)* 27 to 29 labelled a_1(1)* 29 to 7 labelled b_1(1)* 31 to 33 labelled a_1(1)* 33 to 34 labelled a_1(1)* 34 to 8 labelled b_1(1)* 37 to 38 labelled a_1(1)* 37 to 175 labelled c_1(2)* 37 to 183 labelled b_1(3)* 38 to 40 labelled a_1(1)* 38 to 43 labelled c_1(1)* 38 to 51 labelled b_1(2)* 40 to 10 labelled b_1(1)* 43 to 44 labelled c_1(1)* 43 to 54 labelled b_1(2)* 44 to 45 labelled c_1(1)* 44 to 57 labelled b_1(2)* 45 to 46 labelled c_1(1)* 45 to 60 labelled b_1(2)* 46 to 2 labelled a_1(1)* 46 to 43 labelled c_1(1)* 46 to 51 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)* 54 to 55 labelled a_1(2)* 55 to 56 labelled a_1(2)* 56 to 44 labelled b_1(2)* 57 to 58 labelled a_1(2)* 58 to 59 labelled a_1(2)* 59 to 45 labelled b_1(2)* 60 to 61 labelled a_1(2)* 60 to 207 labelled c_1(3)* 60 to 219 labelled b_1(4)* 61 to 62 labelled a_1(2)* 61 to 43 labelled c_1(1)* 61 to 51 labelled b_1(2)* 62 to 46 labelled b_1(2)* 159 to 160 labelled c_1(1)* 159 to 166 labelled b_1(2)* 160 to 161 labelled c_1(1)* 160 to 169 labelled b_1(2)* 161 to 162 labelled c_1(1)* 161 to 172 labelled b_1(2)* 162 to 43 labelled a_1(1)* 162 to 179 labelled c_1(2)* 162 to 186 labelled b_1(3)* 163 to 164 labelled a_1(2)* 164 to 165 labelled a_1(2)* 165 to 159 labelled b_1(2)* 166 to 167 labelled a_1(2)* 167 to 168 labelled a_1(2)* 168 to 160 labelled b_1(2)* 169 to 170 labelled a_1(2)* 170 to 171 labelled a_1(2)* 171 to 161 labelled b_1(2)* 172 to 173 labelled a_1(2)* 173 to 174 labelled a_1(2)* 174 to 162 labelled b_1(2)* 175 to 176 labelled c_1(2)* 175 to 189 labelled b_1(3)* 176 to 177 labelled c_1(2)* 176 to 192 labelled b_1(3)* 177 to 178 labelled c_1(2)* 177 to 195 labelled b_1(3)* 178 to 43 labelled a_1(2)* 178 to 211 labelled c_1(3)* 178 to 222 labelled b_1(4)* 179 to 180 labelled c_1(2)* 179 to 198 labelled b_1(3)* 180 to 181 labelled c_1(2)* 180 to 201 labelled b_1(3)* 181 to 182 labelled c_1(2)* 181 to 204 labelled b_1(3)* 182 to 44 labelled a_1(2)* 182 to 215 labelled c_1(3)* 182 to 225 labelled b_1(4)* 183 to 184 labelled a_1(3)* 184 to 185 labelled a_1(3)* 185 to 175 labelled b_1(3)* 186 to 187 labelled a_1(3)* 187 to 188 labelled a_1(3)* 188 to 179 labelled b_1(3)* 189 to 190 labelled a_1(3)* 190 to 191 labelled a_1(3)* 191 to 176 labelled b_1(3)* 192 to 193 labelled a_1(3)* 193 to 194 labelled a_1(3)* 194 to 177 labelled b_1(3)* 195 to 196 labelled a_1(3)* 196 to 197 labelled a_1(3)* 197 to 178 labelled b_1(3)* 198 to 199 labelled a_1(3)* 199 to 200 labelled a_1(3)* 200 to 180 labelled b_1(3)* 201 to 202 labelled a_1(3)* 202 to 203 labelled a_1(3)* 203 to 181 labelled b_1(3)* 204 to 205 labelled a_1(3)* 205 to 206 labelled a_1(3)* 206 to 182 labelled b_1(3)* 207 to 208 labelled c_1(3)* 207 to 228 labelled b_1(4)* 208 to 209 labelled c_1(3)* 208 to 231 labelled b_1(4)* 209 to 210 labelled c_1(3)* 209 to 234 labelled b_1(4)* 210 to 43 labelled a_1(3)* 210 to 211 labelled c_1(3)* 210 to 222 labelled b_1(4)* 211 to 212 labelled c_1(3)* 211 to 237 labelled b_1(4)* 212 to 213 labelled c_1(3)* 212 to 240 labelled b_1(4)* 213 to 214 labelled c_1(3)* 213 to 243 labelled b_1(4)* 214 to 44 labelled a_1(3)* 214 to 215 labelled c_1(3)* 214 to 225 labelled b_1(4)* 215 to 216 labelled c_1(3)* 215 to 246 labelled b_1(4)* 216 to 217 labelled c_1(3)* 216 to 249 labelled b_1(4)* 217 to 218 labelled c_1(3)* 217 to 252 labelled b_1(4)* 218 to 45 labelled a_1(3)* 218 to 255 labelled c_1(3)* 218 to 259 labelled b_1(4)* 219 to 220 labelled a_1(4)* 220 to 221 labelled a_1(4)* 221 to 207 labelled b_1(4)* 222 to 223 labelled a_1(4)* 223 to 224 labelled a_1(4)* 224 to 211 labelled b_1(4)* 225 to 226 labelled a_1(4)* 226 to 227 labelled a_1(4)* 227 to 215 labelled b_1(4)* 228 to 229 labelled a_1(4)* 229 to 230 labelled a_1(4)* 230 to 208 labelled b_1(4)* 231 to 232 labelled a_1(4)* 232 to 233 labelled a_1(4)* 233 to 209 labelled b_1(4)* 234 to 235 labelled a_1(4)* 235 to 236 labelled a_1(4)* 236 to 210 labelled b_1(4)* 237 to 238 labelled a_1(4)* 238 to 239 labelled a_1(4)* 239 to 212 labelled b_1(4)* 240 to 241 labelled a_1(4)* 241 to 242 labelled a_1(4)* 242 to 213 labelled b_1(4)* 243 to 244 labelled a_1(4)* 244 to 245 labelled a_1(4)* 245 to 214 labelled b_1(4)* 246 to 247 labelled a_1(4)* 247 to 248 labelled a_1(4)* 248 to 216 labelled b_1(4)* 249 to 250 labelled a_1(4)* 250 to 251 labelled a_1(4)* 251 to 217 labelled b_1(4)* 252 to 253 labelled a_1(4)* 253 to 254 labelled a_1(4)* 254 to 218 labelled b_1(4)* 255 to 256 labelled c_1(3)* 255 to 262 labelled b_1(4)* 256 to 257 labelled c_1(3)* 256 to 265 labelled b_1(4)* 257 to 258 labelled c_1(3)* 257 to 268 labelled b_1(4)* 258 to 46 labelled a_1(3)* 258 to 207 labelled c_1(3)* 258 to 219 labelled b_1(4)* 259 to 260 labelled a_1(4)* 260 to 261 labelled a_1(4)* 261 to 255 labelled b_1(4)* 262 to 263 labelled a_1(4)* 263 to 264 labelled a_1(4)* 264 to 256 labelled b_1(4)* 265 to 266 labelled a_1(4)* 266 to 267 labelled a_1(4)* 267 to 257 labelled b_1(4)* 268 to 269 labelled a_1(4)* 268 to 207 labelled c_1(3)* 268 to 219 labelled b_1(4)* 269 to 270 labelled a_1(4)* 269 to 43 labelled c_1(1)* 269 to 51 labelled b_1(2)* 270 to 258 labelled b_1(4) ---------------------------------------- (2) YES