4.54/2.00 YES 4.54/2.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.54/2.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.54/2.02 4.54/2.02 4.54/2.02 Termination of the given RelTRS could be proven: 4.54/2.02 4.54/2.02 (0) RelTRS 4.54/2.02 (1) FlatCCProof [EQUIVALENT, 0 ms] 4.54/2.02 (2) RelTRS 4.54/2.02 (3) RootLabelingProof [EQUIVALENT, 0 ms] 4.54/2.02 (4) RelTRS 4.54/2.02 (5) RelTRSRRRProof [EQUIVALENT, 12 ms] 4.54/2.02 (6) RelTRS 4.54/2.02 (7) RIsEmptyProof [EQUIVALENT, 0 ms] 4.54/2.02 (8) YES 4.54/2.02 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (0) 4.54/2.02 Obligation: 4.54/2.02 Relative term rewrite system: 4.54/2.02 The relative TRS consists of the following R rules: 4.54/2.02 4.54/2.02 a(a(a(x1))) -> c(b(b(x1))) 4.54/2.02 a(a(a(x1))) -> a(c(a(x1))) 4.54/2.02 c(c(b(x1))) -> c(a(b(x1))) 4.54/2.02 4.54/2.02 The relative TRS consists of the following S rules: 4.54/2.02 4.54/2.02 b(c(a(x1))) -> c(b(a(x1))) 4.54/2.02 a(c(c(x1))) -> a(b(c(x1))) 4.54/2.02 a(b(a(x1))) -> b(a(a(x1))) 4.54/2.02 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (1) FlatCCProof (EQUIVALENT) 4.54/2.02 We used flat context closure [ROOTLAB] 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (2) 4.54/2.02 Obligation: 4.54/2.02 Relative term rewrite system: 4.54/2.02 The relative TRS consists of the following R rules: 4.54/2.02 4.54/2.02 a(a(a(x1))) -> a(c(a(x1))) 4.54/2.02 c(c(b(x1))) -> c(a(b(x1))) 4.54/2.02 a(a(a(a(x1)))) -> a(c(b(b(x1)))) 4.54/2.02 c(a(a(a(x1)))) -> c(c(b(b(x1)))) 4.54/2.02 b(a(a(a(x1)))) -> b(c(b(b(x1)))) 4.54/2.02 4.54/2.02 The relative TRS consists of the following S rules: 4.54/2.02 4.54/2.02 a(c(c(x1))) -> a(b(c(x1))) 4.54/2.02 a(b(c(a(x1)))) -> a(c(b(a(x1)))) 4.54/2.02 c(b(c(a(x1)))) -> c(c(b(a(x1)))) 4.54/2.02 b(b(c(a(x1)))) -> b(c(b(a(x1)))) 4.54/2.02 a(a(b(a(x1)))) -> a(b(a(a(x1)))) 4.54/2.02 c(a(b(a(x1)))) -> c(b(a(a(x1)))) 4.54/2.02 b(a(b(a(x1)))) -> b(b(a(a(x1)))) 4.54/2.02 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (3) RootLabelingProof (EQUIVALENT) 4.54/2.02 We used plain root labeling [ROOTLAB] with the following heuristic: 4.54/2.02 LabelAll: All function symbols get labeled 4.54/2.02 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (4) 4.54/2.02 Obligation: 4.54/2.02 Relative term rewrite system: 4.54/2.02 The relative TRS consists of the following R rules: 4.54/2.02 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 4.54/2.02 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 4.54/2.02 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 4.54/2.02 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.54/2.02 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.54/2.02 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.54/2.02 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.54/2.02 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.54/2.02 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.54/2.02 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.54/2.02 4.54/2.02 The relative TRS consists of the following S rules: 4.54/2.02 4.54/2.02 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1))) 4.54/2.02 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1))) 4.54/2.02 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 4.54/2.02 a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 4.54/2.02 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 4.54/2.02 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 4.54/2.02 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 4.54/2.02 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 4.54/2.02 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 4.54/2.02 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 4.54/2.02 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 4.54/2.02 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 4.54/2.02 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.54/2.02 a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.54/2.02 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.54/2.02 c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.54/2.02 c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.54/2.02 c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.54/2.02 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.54/2.02 b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.54/2.02 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.54/2.02 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (5) RelTRSRRRProof (EQUIVALENT) 4.54/2.02 We used the following monotonic ordering for rule removal: 4.54/2.02 Knuth-Bendix order [KBO] with precedence:b_{c_1}_1 > a_{c_1}_1 > c_{c_1}_1 > c_{a_1}_1 > c_{b_1}_1 > b_{b_1}_1 > a_{a_1}_1 > a_{b_1}_1 > b_{a_1}_1 4.54/2.02 4.54/2.02 and weight map: 4.54/2.02 4.54/2.02 a_{a_1}_1=15 4.54/2.02 a_{c_1}_1=1 4.54/2.02 c_{a_1}_1=28 4.54/2.02 a_{b_1}_1=1 4.54/2.02 c_{c_1}_1=18 4.54/2.02 c_{b_1}_1=12 4.54/2.02 b_{a_1}_1=25 4.54/2.02 b_{c_1}_1=16 4.54/2.02 b_{b_1}_1=10 4.54/2.02 4.54/2.02 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 4.54/2.02 Rules from R: 4.54/2.02 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 4.54/2.02 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 4.54/2.02 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 4.54/2.02 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.54/2.02 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.54/2.02 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.54/2.02 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.54/2.02 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.54/2.02 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.54/2.02 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.54/2.02 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.54/2.02 Rules from S: 4.54/2.02 4.54/2.02 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1))) 4.54/2.02 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1))) 4.54/2.02 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 4.54/2.02 a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 4.54/2.02 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 4.54/2.02 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 4.54/2.02 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 4.54/2.02 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 4.54/2.02 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 4.54/2.02 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 4.54/2.02 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 4.54/2.02 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 4.54/2.02 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.54/2.02 a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.54/2.02 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.54/2.02 c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.54/2.02 c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.54/2.02 c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.54/2.02 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.54/2.02 b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.54/2.02 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.54/2.02 4.54/2.02 4.54/2.02 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (6) 4.54/2.02 Obligation: 4.54/2.02 Relative term rewrite system: 4.54/2.02 R is empty. 4.54/2.02 S is empty. 4.54/2.02 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (7) RIsEmptyProof (EQUIVALENT) 4.54/2.02 The TRS R is empty. Hence, termination is trivially proven. 4.54/2.02 ---------------------------------------- 4.54/2.02 4.54/2.02 (8) 4.54/2.02 YES 4.90/2.06 EOF