5.44/2.38 YES 5.44/2.42 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.44/2.42 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.44/2.42 5.44/2.42 5.44/2.42 Termination of the given RelTRS could be proven: 5.44/2.42 5.44/2.42 (0) RelTRS 5.44/2.42 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 5.44/2.42 (2) RelTRS 5.44/2.42 (3) FlatCCProof [EQUIVALENT, 0 ms] 5.44/2.42 (4) RelTRS 5.44/2.42 (5) RootLabelingProof [EQUIVALENT, 0 ms] 5.44/2.42 (6) RelTRS 5.44/2.42 (7) RelTRSRRRProof [EQUIVALENT, 11 ms] 5.44/2.42 (8) RelTRS 5.44/2.42 (9) RIsEmptyProof [EQUIVALENT, 1 ms] 5.44/2.42 (10) YES 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (0) 5.44/2.42 Obligation: 5.44/2.42 Relative term rewrite system: 5.44/2.42 The relative TRS consists of the following R rules: 5.44/2.42 5.44/2.42 a(a(c(x1))) -> a(c(c(x1))) 5.44/2.42 a(a(c(x1))) -> b(a(c(x1))) 5.44/2.42 a(a(c(x1))) -> a(b(c(x1))) 5.44/2.42 c(c(c(x1))) -> b(b(b(x1))) 5.44/2.42 c(a(a(x1))) -> c(b(b(x1))) 5.44/2.42 c(a(c(x1))) -> c(b(c(x1))) 5.44/2.42 5.44/2.42 The relative TRS consists of the following S rules: 5.44/2.42 5.44/2.42 b(b(a(x1))) -> c(c(a(x1))) 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (1) RelTRS Reverse (EQUIVALENT) 5.44/2.42 We have reversed the following relative TRS [REVERSE]: 5.44/2.42 The set of rules R is 5.44/2.42 a(a(c(x1))) -> a(c(c(x1))) 5.44/2.42 a(a(c(x1))) -> b(a(c(x1))) 5.44/2.42 a(a(c(x1))) -> a(b(c(x1))) 5.44/2.42 c(c(c(x1))) -> b(b(b(x1))) 5.44/2.42 c(a(a(x1))) -> c(b(b(x1))) 5.44/2.42 c(a(c(x1))) -> c(b(c(x1))) 5.44/2.42 5.44/2.42 The set of rules S is 5.44/2.42 b(b(a(x1))) -> c(c(a(x1))) 5.44/2.42 5.44/2.42 We have obtained the following relative TRS: 5.44/2.42 The set of rules R is 5.44/2.42 c(a(a(x1))) -> c(c(a(x1))) 5.44/2.42 c(a(a(x1))) -> c(a(b(x1))) 5.44/2.42 c(a(a(x1))) -> c(b(a(x1))) 5.44/2.42 c(c(c(x1))) -> b(b(b(x1))) 5.44/2.42 a(a(c(x1))) -> b(b(c(x1))) 5.44/2.42 c(a(c(x1))) -> c(b(c(x1))) 5.44/2.42 5.44/2.42 The set of rules S is 5.44/2.42 a(b(b(x1))) -> a(c(c(x1))) 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (2) 5.44/2.42 Obligation: 5.44/2.42 Relative term rewrite system: 5.44/2.42 The relative TRS consists of the following R rules: 5.44/2.42 5.44/2.42 c(a(a(x1))) -> c(c(a(x1))) 5.44/2.42 c(a(a(x1))) -> c(a(b(x1))) 5.44/2.42 c(a(a(x1))) -> c(b(a(x1))) 5.44/2.42 c(c(c(x1))) -> b(b(b(x1))) 5.44/2.42 a(a(c(x1))) -> b(b(c(x1))) 5.44/2.42 c(a(c(x1))) -> c(b(c(x1))) 5.44/2.42 5.44/2.42 The relative TRS consists of the following S rules: 5.44/2.42 5.44/2.42 a(b(b(x1))) -> a(c(c(x1))) 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (3) FlatCCProof (EQUIVALENT) 5.44/2.42 We used flat context closure [ROOTLAB] 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (4) 5.44/2.42 Obligation: 5.44/2.42 Relative term rewrite system: 5.44/2.42 The relative TRS consists of the following R rules: 5.44/2.42 5.44/2.42 c(a(a(x1))) -> c(c(a(x1))) 5.44/2.42 c(a(a(x1))) -> c(a(b(x1))) 5.44/2.42 c(a(a(x1))) -> c(b(a(x1))) 5.44/2.42 c(a(c(x1))) -> c(b(c(x1))) 5.44/2.42 c(c(c(c(x1)))) -> c(b(b(b(x1)))) 5.44/2.42 a(c(c(c(x1)))) -> a(b(b(b(x1)))) 5.44/2.42 b(c(c(c(x1)))) -> b(b(b(b(x1)))) 5.44/2.42 c(a(a(c(x1)))) -> c(b(b(c(x1)))) 5.44/2.42 a(a(a(c(x1)))) -> a(b(b(c(x1)))) 5.44/2.42 b(a(a(c(x1)))) -> b(b(b(c(x1)))) 5.44/2.42 5.44/2.42 The relative TRS consists of the following S rules: 5.44/2.42 5.44/2.42 a(b(b(x1))) -> a(c(c(x1))) 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (5) RootLabelingProof (EQUIVALENT) 5.44/2.42 We used plain root labeling [ROOTLAB] with the following heuristic: 5.44/2.42 LabelAll: All function symbols get labeled 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (6) 5.44/2.42 Obligation: 5.44/2.42 Relative term rewrite system: 5.44/2.42 The relative TRS consists of the following R rules: 5.44/2.42 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1))) 5.44/2.42 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{c_1}(c_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 5.44/2.42 c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 5.44/2.42 c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 5.44/2.42 c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 5.44/2.42 a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 5.44/2.42 a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 5.44/2.42 a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 5.44/2.42 b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 5.44/2.42 b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 5.44/2.42 b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 5.44/2.42 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 5.44/2.42 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 5.44/2.42 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 5.44/2.42 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 5.44/2.42 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 5.44/2.42 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 5.44/2.42 5.44/2.42 The relative TRS consists of the following S rules: 5.44/2.42 5.44/2.42 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1))) 5.44/2.42 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1))) 5.44/2.42 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1))) 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (7) RelTRSRRRProof (EQUIVALENT) 5.44/2.42 We used the following monotonic ordering for rule removal: 5.44/2.42 Knuth-Bendix order [KBO] with precedence:c_{c_1}_1 > c_{a_1}_1 > c_{b_1}_1 > a_{b_1}_1 > a_{c_1}_1 > b_{c_1}_1 > b_{a_1}_1 > a_{a_1}_1 > b_{b_1}_1 5.44/2.42 5.44/2.42 and weight map: 5.44/2.42 5.44/2.42 c_{a_1}_1=20 5.44/2.42 a_{a_1}_1=19 5.44/2.42 a_{c_1}_1=1 5.44/2.42 c_{c_1}_1=14 5.44/2.42 a_{b_1}_1=3 5.44/2.42 b_{c_1}_1=16 5.44/2.42 b_{a_1}_1=23 5.44/2.42 b_{b_1}_1=10 5.44/2.42 c_{b_1}_1=5 5.44/2.42 5.44/2.42 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 5.44/2.42 Rules from R: 5.44/2.42 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1))) 5.44/2.42 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(x1))) 5.44/2.42 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{c_1}(c_{a_1}(x1))) 5.44/2.42 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 5.44/2.42 c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 5.44/2.42 c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 5.44/2.42 c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 5.44/2.42 a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 5.44/2.42 a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 5.44/2.42 a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 5.44/2.42 b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 5.44/2.42 b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 5.44/2.42 b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 5.44/2.42 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 5.44/2.42 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 5.44/2.42 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 5.44/2.42 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 5.44/2.42 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 5.44/2.42 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 5.44/2.42 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 5.44/2.42 Rules from S: 5.44/2.42 5.44/2.42 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1))) 5.44/2.42 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1))) 5.44/2.42 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1))) 5.44/2.42 5.44/2.42 5.44/2.42 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (8) 5.44/2.42 Obligation: 5.44/2.42 Relative term rewrite system: 5.44/2.42 R is empty. 5.44/2.42 S is empty. 5.44/2.42 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (9) RIsEmptyProof (EQUIVALENT) 5.44/2.42 The TRS R is empty. Hence, termination is trivially proven. 5.44/2.42 ---------------------------------------- 5.44/2.42 5.44/2.42 (10) 5.44/2.42 YES 5.77/2.47 EOF