6.79/2.49 YES 6.79/2.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 6.79/2.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.79/2.51 6.79/2.51 6.79/2.51 Termination of the given RelTRS could be proven: 6.79/2.51 6.79/2.51 (0) RelTRS 6.79/2.51 (1) FlatCCProof [EQUIVALENT, 0 ms] 6.79/2.51 (2) RelTRS 6.79/2.51 (3) RootLabelingProof [EQUIVALENT, 8 ms] 6.79/2.51 (4) RelTRS 6.79/2.51 (5) RelTRSRRRProof [EQUIVALENT, 19 ms] 6.79/2.51 (6) RelTRS 6.79/2.51 (7) RIsEmptyProof [EQUIVALENT, 5 ms] 6.79/2.51 (8) YES 6.79/2.51 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (0) 6.79/2.51 Obligation: 6.79/2.51 Relative term rewrite system: 6.79/2.51 The relative TRS consists of the following R rules: 6.79/2.51 6.79/2.51 a(a(c(x1))) -> c(c(c(x1))) 6.79/2.51 a(c(b(x1))) -> a(a(c(x1))) 6.79/2.51 b(c(a(x1))) -> b(a(b(x1))) 6.79/2.51 a(b(c(x1))) -> b(a(c(x1))) 6.79/2.51 6.79/2.51 The relative TRS consists of the following S rules: 6.79/2.51 6.79/2.51 b(c(b(x1))) -> a(b(c(x1))) 6.79/2.51 a(c(b(x1))) -> c(c(b(x1))) 6.79/2.51 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (1) FlatCCProof (EQUIVALENT) 6.79/2.51 We used flat context closure [ROOTLAB] 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (2) 6.79/2.51 Obligation: 6.79/2.51 Relative term rewrite system: 6.79/2.51 The relative TRS consists of the following R rules: 6.79/2.51 6.79/2.51 a(c(b(x1))) -> a(a(c(x1))) 6.79/2.51 b(c(a(x1))) -> b(a(b(x1))) 6.79/2.51 a(a(a(c(x1)))) -> a(c(c(c(x1)))) 6.79/2.51 c(a(a(c(x1)))) -> c(c(c(c(x1)))) 6.79/2.51 b(a(a(c(x1)))) -> b(c(c(c(x1)))) 6.79/2.51 a(a(b(c(x1)))) -> a(b(a(c(x1)))) 6.79/2.51 c(a(b(c(x1)))) -> c(b(a(c(x1)))) 6.79/2.51 b(a(b(c(x1)))) -> b(b(a(c(x1)))) 6.79/2.51 6.79/2.51 The relative TRS consists of the following S rules: 6.79/2.51 6.79/2.51 a(b(c(b(x1)))) -> a(a(b(c(x1)))) 6.79/2.51 c(b(c(b(x1)))) -> c(a(b(c(x1)))) 6.79/2.51 b(b(c(b(x1)))) -> b(a(b(c(x1)))) 6.79/2.51 a(a(c(b(x1)))) -> a(c(c(b(x1)))) 6.79/2.51 c(a(c(b(x1)))) -> c(c(c(b(x1)))) 6.79/2.51 b(a(c(b(x1)))) -> b(c(c(b(x1)))) 6.79/2.51 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (3) RootLabelingProof (EQUIVALENT) 6.79/2.51 We used plain root labeling [ROOTLAB] with the following heuristic: 6.79/2.51 LabelAll: All function symbols get labeled 6.79/2.51 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (4) 6.79/2.51 Obligation: 6.79/2.51 Relative term rewrite system: 6.79/2.51 The relative TRS consists of the following R rules: 6.79/2.51 6.79/2.51 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1))) 6.79/2.51 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1))) 6.79/2.51 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1))) 6.79/2.51 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1))) 6.79/2.51 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(x1))) 6.79/2.51 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1))) 6.79/2.51 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 6.79/2.51 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 6.79/2.51 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 6.79/2.51 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 6.79/2.51 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 6.79/2.51 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 6.79/2.51 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 6.79/2.51 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 6.79/2.51 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 6.79/2.51 a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) 6.79/2.51 a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) 6.79/2.51 a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) 6.79/2.51 c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) 6.79/2.51 c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) 6.79/2.51 c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) 6.79/2.51 b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) 6.79/2.51 b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) 6.79/2.51 b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) 6.79/2.51 6.79/2.51 The relative TRS consists of the following S rules: 6.79/2.51 6.79/2.51 a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 6.79/2.51 a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 6.79/2.51 a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 6.79/2.51 c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 6.79/2.51 c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 6.79/2.51 c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 6.79/2.51 b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 6.79/2.51 b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 6.79/2.51 b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 6.79/2.51 a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 6.79/2.51 a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 6.79/2.51 a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 6.79/2.51 c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 6.79/2.51 c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 6.79/2.51 c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 6.79/2.51 b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 6.79/2.51 b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 6.79/2.51 b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 6.79/2.51 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (5) RelTRSRRRProof (EQUIVALENT) 6.79/2.51 We used the following monotonic ordering for rule removal: 6.79/2.51 Knuth-Bendix order [KBO] with precedence:b_{b_1}_1 > c_{c_1}_1 > a_{a_1}_1 > a_{b_1}_1 > c_{a_1}_1 > b_{a_1}_1 > b_{c_1}_1 > c_{b_1}_1 > a_{c_1}_1 6.79/2.51 6.79/2.51 and weight map: 6.79/2.51 6.79/2.51 a_{c_1}_1=1 6.79/2.51 c_{b_1}_1=5 6.79/2.51 b_{a_1}_1=2 6.79/2.51 a_{a_1}_1=1 6.79/2.51 c_{a_1}_1=4 6.79/2.51 b_{c_1}_1=2 6.79/2.51 c_{c_1}_1=1 6.79/2.51 b_{b_1}_1=2 6.79/2.51 a_{b_1}_1=2 6.79/2.51 6.79/2.51 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 6.79/2.51 Rules from R: 6.79/2.51 6.79/2.51 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1))) 6.79/2.51 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1))) 6.79/2.51 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1))) 6.79/2.51 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1))) 6.79/2.51 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(x1))) 6.79/2.51 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1))) 6.79/2.51 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 6.79/2.51 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 6.79/2.51 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 6.79/2.51 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 6.79/2.51 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 6.79/2.51 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 6.79/2.51 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 6.79/2.51 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 6.79/2.51 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 6.79/2.51 a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) 6.79/2.51 a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) 6.79/2.51 a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) 6.79/2.51 c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) 6.79/2.51 c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) 6.79/2.51 c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) 6.79/2.51 b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) 6.79/2.51 b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) 6.79/2.51 b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) 6.79/2.51 Rules from S: 6.79/2.51 6.79/2.51 a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 6.79/2.51 a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 6.79/2.51 a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 6.79/2.51 c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 6.79/2.51 c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 6.79/2.51 c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 6.79/2.51 b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 6.79/2.51 b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 6.79/2.51 b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 6.79/2.51 a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 6.79/2.51 a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 6.79/2.51 a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 6.79/2.51 c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 6.79/2.51 c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 6.79/2.51 c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 6.79/2.51 b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 6.79/2.51 b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 6.79/2.51 b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 6.79/2.51 6.79/2.51 6.79/2.51 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (6) 6.79/2.51 Obligation: 6.79/2.51 Relative term rewrite system: 6.79/2.51 R is empty. 6.79/2.51 S is empty. 6.79/2.51 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (7) RIsEmptyProof (EQUIVALENT) 6.79/2.51 The TRS R is empty. Hence, termination is trivially proven. 6.79/2.51 ---------------------------------------- 6.79/2.51 6.79/2.51 (8) 6.79/2.51 YES 6.79/2.54 EOF