13.44/4.25 YES 13.70/4.33 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 13.70/4.33 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 13.70/4.33 13.70/4.33 13.70/4.33 Termination of the given RelTRS could be proven: 13.70/4.33 13.70/4.33 (0) RelTRS 13.70/4.33 (1) RelTRSRRRProof [EQUIVALENT, 56 ms] 13.70/4.33 (2) RelTRS 13.70/4.33 (3) SIsEmptyProof [EQUIVALENT, 0 ms] 13.70/4.33 (4) QTRS 13.70/4.33 (5) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] 13.70/4.33 (6) YES 13.70/4.33 13.70/4.33 13.70/4.33 ---------------------------------------- 13.70/4.33 13.70/4.33 (0) 13.70/4.33 Obligation: 13.70/4.33 Relative term rewrite system: 13.70/4.33 The relative TRS consists of the following R rules: 13.70/4.33 13.70/4.33 c(a(b(x1))) -> a(b(c(x1))) 13.70/4.33 a(b(a(x1))) -> c(a(a(x1))) 13.70/4.33 c(c(c(x1))) -> b(c(c(x1))) 13.70/4.33 b(b(a(x1))) -> b(b(b(x1))) 13.70/4.33 13.70/4.33 The relative TRS consists of the following S rules: 13.70/4.33 13.70/4.33 a(c(b(x1))) -> c(b(c(x1))) 13.70/4.33 a(c(b(x1))) -> b(b(c(x1))) 13.70/4.33 13.70/4.33 13.70/4.33 ---------------------------------------- 13.70/4.33 13.70/4.33 (1) RelTRSRRRProof (EQUIVALENT) 13.70/4.33 We used the following monotonic ordering for rule removal: 13.70/4.33 Polynomial interpretation [POLO]: 13.70/4.33 13.70/4.33 POL(a(x_1)) = 1 + x_1 13.70/4.33 POL(b(x_1)) = x_1 13.70/4.33 POL(c(x_1)) = x_1 13.70/4.33 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 13.70/4.33 Rules from R: 13.70/4.33 13.70/4.33 b(b(a(x1))) -> b(b(b(x1))) 13.70/4.33 Rules from S: 13.70/4.33 13.70/4.33 a(c(b(x1))) -> c(b(c(x1))) 13.70/4.33 a(c(b(x1))) -> b(b(c(x1))) 13.70/4.33 13.70/4.33 13.70/4.33 13.70/4.33 13.70/4.33 ---------------------------------------- 13.70/4.33 13.70/4.33 (2) 13.70/4.33 Obligation: 13.70/4.33 Relative term rewrite system: 13.70/4.33 The relative TRS consists of the following R rules: 13.70/4.33 13.70/4.33 c(a(b(x1))) -> a(b(c(x1))) 13.70/4.33 a(b(a(x1))) -> c(a(a(x1))) 13.70/4.33 c(c(c(x1))) -> b(c(c(x1))) 13.70/4.33 13.70/4.33 S is empty. 13.70/4.33 13.70/4.33 ---------------------------------------- 13.70/4.33 13.70/4.33 (3) SIsEmptyProof (EQUIVALENT) 13.70/4.33 The TRS S is empty. Hence, termination of R/S is equivalent to termination of R. 13.70/4.33 ---------------------------------------- 13.70/4.33 13.70/4.33 (4) 13.70/4.33 Obligation: 13.70/4.33 Q restricted rewrite system: 13.70/4.33 The TRS R consists of the following rules: 13.70/4.33 13.70/4.33 c(a(b(x1))) -> a(b(c(x1))) 13.70/4.33 a(b(a(x1))) -> c(a(a(x1))) 13.70/4.33 c(c(c(x1))) -> b(c(c(x1))) 13.70/4.33 13.70/4.33 Q is empty. 13.70/4.33 13.70/4.33 ---------------------------------------- 13.70/4.33 13.70/4.33 (5) RFCMatchBoundsTRSProof (EQUIVALENT) 13.70/4.33 Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. This implies Q-termination of R. 13.70/4.33 The following rules were used to construct the certificate: 13.70/4.33 13.70/4.33 c(a(b(x1))) -> a(b(c(x1))) 13.70/4.33 a(b(a(x1))) -> c(a(a(x1))) 13.70/4.33 c(c(c(x1))) -> b(c(c(x1))) 13.70/4.33 13.70/4.33 The certificate found is represented by the following graph. 13.70/4.33 The certificate consists of the following enumerated nodes: 13.70/4.33 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 165, 166, 167, 168, 169, 170, 171, 172 13.70/4.33 13.70/4.33 Node 129 is start node and node 130 is final node. 13.70/4.33 13.70/4.33 Those nodes are connected through the following edges: 13.70/4.33 13.70/4.33 * 129 to 131 labelled a_1(0)* 129 to 133 labelled c_1(0)* 129 to 135 labelled b_1(0)* 129 to 143 labelled c_1(1)* 130 to 130 labelled #_1(0)* 131 to 132 labelled b_1(0)* 132 to 130 labelled c_1(0)* 132 to 137 labelled a_1(1)* 132 to 139 labelled b_1(1)* 132 to 165 labelled c_1(2)* 133 to 134 labelled a_1(0)* 134 to 130 labelled a_1(0)* 134 to 141 labelled c_1(1)* 135 to 136 labelled c_1(0)* 135 to 139 labelled b_1(1)* 135 to 145 labelled a_1(1)* 135 to 169 labelled c_1(2)* 136 to 130 labelled c_1(0)* 136 to 137 labelled a_1(1)* 136 to 139 labelled b_1(1)* 136 to 165 labelled c_1(2)* 137 to 138 labelled b_1(1)* 138 to 130 labelled c_1(1)* 138 to 137 labelled a_1(1)* 138 to 139 labelled b_1(1)* 138 to 165 labelled c_1(2)* 139 to 140 labelled c_1(1)* 139 to 139 labelled b_1(1)* 139 to 167 labelled a_1(2)* 139 to 171 labelled c_1(3)* 140 to 130 labelled c_1(1)* 140 to 137 labelled a_1(1)* 140 to 139 labelled b_1(1)* 140 to 165 labelled c_1(2)* 141 to 142 labelled a_1(1)* 142 to 130 labelled a_1(1)* 142 to 141 labelled c_1(1)* 143 to 144 labelled a_1(1)* 144 to 137 labelled a_1(1)* 144 to 165 labelled c_1(2)* 145 to 146 labelled b_1(1)* 146 to 138 labelled c_1(1)* 146 to 146 labelled b_1(1)* 146 to 167 labelled a_1(2)* 146 to 171 labelled c_1(3)* 165 to 166 labelled a_1(2)* 166 to 137 labelled a_1(2)* 166 to 165 labelled c_1(2)* 167 to 168 labelled b_1(2)* 168 to 138 labelled c_1(2)* 168 to 146 labelled b_1(1)* 168 to 167 labelled a_1(2)* 168 to 171 labelled c_1(3)* 169 to 170 labelled a_1(2)* 170 to 167 labelled a_1(2)* 170 to 171 labelled c_1(3)* 171 to 172 labelled a_1(3)* 172 to 167 labelled a_1(3)* 172 to 171 labelled c_1(3) 13.70/4.33 13.70/4.33 13.70/4.33 ---------------------------------------- 13.70/4.33 13.70/4.33 (6) 13.70/4.33 YES 14.03/4.44 EOF