5.33/2.15 YES 5.33/2.18 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.33/2.18 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.33/2.18 5.33/2.18 5.33/2.18 Termination of the given RelTRS could be proven: 5.33/2.18 5.33/2.18 (0) RelTRS 5.33/2.18 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 5.33/2.18 (2) RelTRS 5.33/2.18 (3) RelTRSRRRProof [EQUIVALENT, 165 ms] 5.33/2.18 (4) RelTRS 5.33/2.18 (5) RIsEmptyProof [EQUIVALENT, 0 ms] 5.33/2.18 (6) YES 5.33/2.18 5.33/2.18 5.33/2.18 ---------------------------------------- 5.33/2.18 5.33/2.18 (0) 5.33/2.18 Obligation: 5.33/2.18 Relative term rewrite system: 5.33/2.18 The relative TRS consists of the following R rules: 5.33/2.18 5.33/2.18 a(b(a(x1))) -> a(b(b(a(x1)))) 5.33/2.18 5.33/2.18 The relative TRS consists of the following S rules: 5.33/2.18 5.33/2.18 b(x1) -> b(b(b(x1))) 5.33/2.18 5.33/2.18 5.33/2.18 ---------------------------------------- 5.33/2.18 5.33/2.18 (1) RelTRS Reverse (EQUIVALENT) 5.33/2.18 We have reversed the following relative TRS [REVERSE]: 5.33/2.18 The set of rules R is 5.33/2.18 a(b(a(x1))) -> a(b(b(a(x1)))) 5.33/2.18 5.33/2.18 The set of rules S is 5.33/2.18 b(x1) -> b(b(b(x1))) 5.33/2.18 5.33/2.18 We have obtained the following relative TRS: 5.33/2.18 The set of rules R is 5.33/2.18 a(b(a(x1))) -> a(b(b(a(x1)))) 5.33/2.18 5.33/2.18 The set of rules S is 5.33/2.18 b(x1) -> b(b(b(x1))) 5.33/2.18 5.33/2.18 5.33/2.18 ---------------------------------------- 5.33/2.18 5.33/2.18 (2) 5.33/2.18 Obligation: 5.33/2.18 Relative term rewrite system: 5.33/2.18 The relative TRS consists of the following R rules: 5.33/2.18 5.33/2.18 a(b(a(x1))) -> a(b(b(a(x1)))) 5.33/2.18 5.33/2.18 The relative TRS consists of the following S rules: 5.33/2.18 5.33/2.18 b(x1) -> b(b(b(x1))) 5.33/2.18 5.33/2.18 5.33/2.18 ---------------------------------------- 5.33/2.18 5.33/2.18 (3) RelTRSRRRProof (EQUIVALENT) 5.33/2.18 We used the following monotonic ordering for rule removal: 5.33/2.18 Matrix interpretation [MATRO] to (N^3, +, *, >=, >) : 5.33/2.18 5.33/2.18 <<< 5.33/2.18 POL(a(x_1)) = [[0], [0], [3]] + [[2, 2, 1], [0, 0, 0], [2, 3, 1]] * x_1 5.33/2.18 >>> 5.33/2.18 5.33/2.18 <<< 5.33/2.18 POL(b(x_1)) = [[0], [2], [0]] + [[1, 0, 0], [0, 0, 2], [0, 0, 0]] * x_1 5.33/2.18 >>> 5.33/2.18 5.33/2.18 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 5.33/2.18 Rules from R: 5.33/2.18 5.33/2.18 a(b(a(x1))) -> a(b(b(a(x1)))) 5.33/2.18 Rules from S: 5.33/2.18 none 5.33/2.18 5.33/2.18 5.33/2.18 5.33/2.18 5.33/2.18 ---------------------------------------- 5.33/2.18 5.33/2.18 (4) 5.33/2.18 Obligation: 5.33/2.18 Relative term rewrite system: 5.33/2.18 R is empty. 5.33/2.18 The relative TRS consists of the following S rules: 5.33/2.18 5.33/2.18 b(x1) -> b(b(b(x1))) 5.33/2.18 5.33/2.18 5.33/2.18 ---------------------------------------- 5.33/2.18 5.33/2.18 (5) RIsEmptyProof (EQUIVALENT) 5.33/2.18 The TRS R is empty. Hence, termination is trivially proven. 5.33/2.18 ---------------------------------------- 5.33/2.18 5.33/2.18 (6) 5.33/2.18 YES 5.64/2.22 EOF