7.70/2.86 YES 7.70/2.89 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 7.70/2.89 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.70/2.89 7.70/2.89 7.70/2.89 Termination of the given RelTRS could be proven: 7.70/2.89 7.70/2.89 (0) RelTRS 7.70/2.89 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 7.70/2.89 (2) RelTRS 7.70/2.89 (3) FlatCCProof [EQUIVALENT, 0 ms] 7.70/2.89 (4) RelTRS 7.70/2.89 (5) RootLabelingProof [EQUIVALENT, 0 ms] 7.70/2.89 (6) RelTRS 7.70/2.89 (7) RelTRSRRRProof [EQUIVALENT, 52 ms] 7.70/2.89 (8) RelTRS 7.70/2.89 (9) RelTRSRRRProof [EQUIVALENT, 4 ms] 7.70/2.89 (10) RelTRS 7.70/2.89 (11) RIsEmptyProof [EQUIVALENT, 6 ms] 7.70/2.89 (12) YES 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (0) 7.70/2.89 Obligation: 7.70/2.89 Relative term rewrite system: 7.70/2.89 The relative TRS consists of the following R rules: 7.70/2.89 7.70/2.89 a(a(b(x1))) -> c(c(c(x1))) 7.70/2.89 a(c(a(x1))) -> c(c(a(x1))) 7.70/2.89 a(c(a(x1))) -> b(a(a(x1))) 7.70/2.89 7.70/2.89 The relative TRS consists of the following S rules: 7.70/2.89 7.70/2.89 b(a(c(x1))) -> b(b(c(x1))) 7.70/2.89 b(c(b(x1))) -> a(b(b(x1))) 7.70/2.89 c(a(b(x1))) -> b(b(a(x1))) 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (1) RelTRS Reverse (EQUIVALENT) 7.70/2.89 We have reversed the following relative TRS [REVERSE]: 7.70/2.89 The set of rules R is 7.70/2.89 a(a(b(x1))) -> c(c(c(x1))) 7.70/2.89 a(c(a(x1))) -> c(c(a(x1))) 7.70/2.89 a(c(a(x1))) -> b(a(a(x1))) 7.70/2.89 7.70/2.89 The set of rules S is 7.70/2.89 b(a(c(x1))) -> b(b(c(x1))) 7.70/2.89 b(c(b(x1))) -> a(b(b(x1))) 7.70/2.89 c(a(b(x1))) -> b(b(a(x1))) 7.70/2.89 7.70/2.89 We have obtained the following relative TRS: 7.70/2.89 The set of rules R is 7.70/2.89 b(a(a(x1))) -> c(c(c(x1))) 7.70/2.89 a(c(a(x1))) -> a(c(c(x1))) 7.70/2.89 a(c(a(x1))) -> a(a(b(x1))) 7.70/2.89 7.70/2.89 The set of rules S is 7.70/2.89 c(a(b(x1))) -> c(b(b(x1))) 7.70/2.89 b(c(b(x1))) -> b(b(a(x1))) 7.70/2.89 b(a(c(x1))) -> a(b(b(x1))) 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (2) 7.70/2.89 Obligation: 7.70/2.89 Relative term rewrite system: 7.70/2.89 The relative TRS consists of the following R rules: 7.70/2.89 7.70/2.89 b(a(a(x1))) -> c(c(c(x1))) 7.70/2.89 a(c(a(x1))) -> a(c(c(x1))) 7.70/2.89 a(c(a(x1))) -> a(a(b(x1))) 7.70/2.89 7.70/2.89 The relative TRS consists of the following S rules: 7.70/2.89 7.70/2.89 c(a(b(x1))) -> c(b(b(x1))) 7.70/2.89 b(c(b(x1))) -> b(b(a(x1))) 7.70/2.89 b(a(c(x1))) -> a(b(b(x1))) 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (3) FlatCCProof (EQUIVALENT) 7.70/2.89 We used flat context closure [ROOTLAB] 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (4) 7.70/2.89 Obligation: 7.70/2.89 Relative term rewrite system: 7.70/2.89 The relative TRS consists of the following R rules: 7.70/2.89 7.70/2.89 a(c(a(x1))) -> a(c(c(x1))) 7.70/2.89 a(c(a(x1))) -> a(a(b(x1))) 7.70/2.89 b(b(a(a(x1)))) -> b(c(c(c(x1)))) 7.70/2.89 a(b(a(a(x1)))) -> a(c(c(c(x1)))) 7.70/2.89 c(b(a(a(x1)))) -> c(c(c(c(x1)))) 7.70/2.89 7.70/2.89 The relative TRS consists of the following S rules: 7.70/2.89 7.70/2.89 c(a(b(x1))) -> c(b(b(x1))) 7.70/2.89 b(c(b(x1))) -> b(b(a(x1))) 7.70/2.89 b(b(a(c(x1)))) -> b(a(b(b(x1)))) 7.70/2.89 a(b(a(c(x1)))) -> a(a(b(b(x1)))) 7.70/2.89 c(b(a(c(x1)))) -> c(a(b(b(x1)))) 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (5) RootLabelingProof (EQUIVALENT) 7.70/2.89 We used plain root labeling [ROOTLAB] with the following heuristic: 7.70/2.89 LabelAll: All function symbols get labeled 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (6) 7.70/2.89 Obligation: 7.70/2.89 Relative term rewrite system: 7.70/2.89 The relative TRS consists of the following R rules: 7.70/2.89 7.70/2.89 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 7.70/2.89 The relative TRS consists of the following S rules: 7.70/2.89 7.70/2.89 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1))) 7.70/2.89 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1))) 7.70/2.89 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1))) 7.70/2.89 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1))) 7.70/2.89 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 7.70/2.89 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (7) RelTRSRRRProof (EQUIVALENT) 7.70/2.89 We used the following monotonic ordering for rule removal: 7.70/2.89 Polynomial interpretation [POLO]: 7.70/2.89 7.70/2.89 POL(a_{a_1}(x_1)) = x_1 7.70/2.89 POL(a_{b_1}(x_1)) = x_1 7.70/2.89 POL(a_{c_1}(x_1)) = x_1 7.70/2.89 POL(b_{a_1}(x_1)) = 1 + x_1 7.70/2.89 POL(b_{b_1}(x_1)) = x_1 7.70/2.89 POL(b_{c_1}(x_1)) = x_1 7.70/2.89 POL(c_{a_1}(x_1)) = 1 + x_1 7.70/2.89 POL(c_{b_1}(x_1)) = 1 + x_1 7.70/2.89 POL(c_{c_1}(x_1)) = x_1 7.70/2.89 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.70/2.89 Rules from R: 7.70/2.89 7.70/2.89 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 Rules from S: 7.70/2.89 7.70/2.89 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 7.70/2.89 c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (8) 7.70/2.89 Obligation: 7.70/2.89 Relative term rewrite system: 7.70/2.89 The relative TRS consists of the following R rules: 7.70/2.89 7.70/2.89 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 7.70/2.89 The relative TRS consists of the following S rules: 7.70/2.89 7.70/2.89 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1))) 7.70/2.89 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1))) 7.70/2.89 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1))) 7.70/2.89 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 7.70/2.89 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (9) RelTRSRRRProof (EQUIVALENT) 7.70/2.89 We used the following monotonic ordering for rule removal: 7.70/2.89 Knuth-Bendix order [KBO] with precedence:c_{b_1}_1 > b_{b_1}_1 > a_{c_1}_1 > b_{a_1}_1 > a_{b_1}_1 > a_{a_1}_1 > b_{c_1}_1 > c_{a_1}_1 > c_{c_1}_1 7.70/2.89 7.70/2.89 and weight map: 7.70/2.89 7.70/2.89 a_{c_1}_1=11 7.70/2.89 c_{a_1}_1=9 7.70/2.89 a_{a_1}_1=22 7.70/2.89 c_{c_1}_1=15 7.70/2.89 a_{b_1}_1=7 7.70/2.89 c_{b_1}_1=1 7.70/2.89 b_{a_1}_1=7 7.70/2.89 b_{b_1}_1=14 7.70/2.89 b_{c_1}_1=16 7.70/2.89 7.70/2.89 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.70/2.89 Rules from R: 7.70/2.89 7.70/2.89 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1))) 7.70/2.89 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 7.70/2.89 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 7.70/2.89 Rules from S: 7.70/2.89 7.70/2.89 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1))) 7.70/2.89 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1))) 7.70/2.89 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1))) 7.70/2.89 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 7.70/2.89 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 7.70/2.89 b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (10) 7.70/2.89 Obligation: 7.70/2.89 Relative term rewrite system: 7.70/2.89 R is empty. 7.70/2.89 S is empty. 7.70/2.89 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (11) RIsEmptyProof (EQUIVALENT) 7.70/2.89 The TRS R is empty. Hence, termination is trivially proven. 7.70/2.89 ---------------------------------------- 7.70/2.89 7.70/2.89 (12) 7.70/2.89 YES 8.00/2.93 EOF