8.85/3.07 YES 8.85/3.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 8.85/3.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 8.85/3.08 8.85/3.08 8.85/3.08 Termination of the given RelTRS could be proven: 8.85/3.08 8.85/3.08 (0) RelTRS 8.85/3.08 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 8.85/3.08 (2) RelTRS 8.85/3.08 (3) RelTRSRRRProof [EQUIVALENT, 173 ms] 8.85/3.08 (4) RelTRS 8.85/3.08 (5) RelTRSRRRProof [EQUIVALENT, 5 ms] 8.85/3.08 (6) RelTRS 8.85/3.08 (7) RelTRSRRRProof [EQUIVALENT, 10 ms] 8.85/3.08 (8) RelTRS 8.85/3.08 (9) RIsEmptyProof [EQUIVALENT, 0 ms] 8.85/3.08 (10) YES 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (0) 8.85/3.08 Obligation: 8.85/3.08 Relative term rewrite system: 8.85/3.08 The relative TRS consists of the following R rules: 8.85/3.08 8.85/3.08 b(c(b(x1))) -> b(a(c(x1))) 8.85/3.08 b(a(c(x1))) -> b(c(a(x1))) 8.85/3.08 8.85/3.08 The relative TRS consists of the following S rules: 8.85/3.08 8.85/3.08 a(c(c(x1))) -> c(c(b(x1))) 8.85/3.08 a(c(a(x1))) -> a(b(b(x1))) 8.85/3.08 b(b(b(x1))) -> c(c(a(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (1) RelTRS Reverse (EQUIVALENT) 8.85/3.08 We have reversed the following relative TRS [REVERSE]: 8.85/3.08 The set of rules R is 8.85/3.08 b(c(b(x1))) -> b(a(c(x1))) 8.85/3.08 b(a(c(x1))) -> b(c(a(x1))) 8.85/3.08 8.85/3.08 The set of rules S is 8.85/3.08 a(c(c(x1))) -> c(c(b(x1))) 8.85/3.08 a(c(a(x1))) -> a(b(b(x1))) 8.85/3.08 b(b(b(x1))) -> c(c(a(x1))) 8.85/3.08 8.85/3.08 We have obtained the following relative TRS: 8.85/3.08 The set of rules R is 8.85/3.08 b(c(b(x1))) -> c(a(b(x1))) 8.85/3.08 c(a(b(x1))) -> a(c(b(x1))) 8.85/3.08 8.85/3.08 The set of rules S is 8.85/3.08 c(c(a(x1))) -> b(c(c(x1))) 8.85/3.08 a(c(a(x1))) -> b(b(a(x1))) 8.85/3.08 b(b(b(x1))) -> a(c(c(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (2) 8.85/3.08 Obligation: 8.85/3.08 Relative term rewrite system: 8.85/3.08 The relative TRS consists of the following R rules: 8.85/3.08 8.85/3.08 b(c(b(x1))) -> c(a(b(x1))) 8.85/3.08 c(a(b(x1))) -> a(c(b(x1))) 8.85/3.08 8.85/3.08 The relative TRS consists of the following S rules: 8.85/3.08 8.85/3.08 c(c(a(x1))) -> b(c(c(x1))) 8.85/3.08 a(c(a(x1))) -> b(b(a(x1))) 8.85/3.08 b(b(b(x1))) -> a(c(c(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (3) RelTRSRRRProof (EQUIVALENT) 8.85/3.08 We used the following monotonic ordering for rule removal: 8.85/3.08 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 8.85/3.08 8.85/3.08 <<< 8.85/3.08 POL(b(x_1)) = [[1], [0]] + [[1, 1], [0, 1]] * x_1 8.85/3.08 >>> 8.85/3.08 8.85/3.08 <<< 8.85/3.08 POL(c(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 8.85/3.08 >>> 8.85/3.08 8.85/3.08 <<< 8.85/3.08 POL(a(x_1)) = [[2], [0]] + [[1, 0], [0, 0]] * x_1 8.85/3.08 >>> 8.85/3.08 8.85/3.08 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 8.85/3.08 Rules from R: 8.85/3.08 none 8.85/3.08 Rules from S: 8.85/3.08 8.85/3.08 b(b(b(x1))) -> a(c(c(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (4) 8.85/3.08 Obligation: 8.85/3.08 Relative term rewrite system: 8.85/3.08 The relative TRS consists of the following R rules: 8.85/3.08 8.85/3.08 b(c(b(x1))) -> c(a(b(x1))) 8.85/3.08 c(a(b(x1))) -> a(c(b(x1))) 8.85/3.08 8.85/3.08 The relative TRS consists of the following S rules: 8.85/3.08 8.85/3.08 c(c(a(x1))) -> b(c(c(x1))) 8.85/3.08 a(c(a(x1))) -> b(b(a(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (5) RelTRSRRRProof (EQUIVALENT) 8.85/3.08 We used the following monotonic ordering for rule removal: 8.85/3.08 Polynomial interpretation [POLO]: 8.85/3.08 8.85/3.08 POL(a(x_1)) = x_1 8.85/3.08 POL(b(x_1)) = x_1 8.85/3.08 POL(c(x_1)) = 1 + x_1 8.85/3.08 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 8.85/3.08 Rules from R: 8.85/3.08 none 8.85/3.08 Rules from S: 8.85/3.08 8.85/3.08 a(c(a(x1))) -> b(b(a(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (6) 8.85/3.08 Obligation: 8.85/3.08 Relative term rewrite system: 8.85/3.08 The relative TRS consists of the following R rules: 8.85/3.08 8.85/3.08 b(c(b(x1))) -> c(a(b(x1))) 8.85/3.08 c(a(b(x1))) -> a(c(b(x1))) 8.85/3.08 8.85/3.08 The relative TRS consists of the following S rules: 8.85/3.08 8.85/3.08 c(c(a(x1))) -> b(c(c(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (7) RelTRSRRRProof (EQUIVALENT) 8.85/3.08 We used the following monotonic ordering for rule removal: 8.85/3.08 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 8.85/3.08 8.85/3.08 <<< 8.85/3.08 POL(b(x_1)) = [[0], [0]] + [[1, 2], [0, 1]] * x_1 8.85/3.08 >>> 8.85/3.08 8.85/3.08 <<< 8.85/3.08 POL(c(x_1)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 8.85/3.08 >>> 8.85/3.08 8.85/3.08 <<< 8.85/3.08 POL(a(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 8.85/3.08 >>> 8.85/3.08 8.85/3.08 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 8.85/3.08 Rules from R: 8.85/3.08 8.85/3.08 b(c(b(x1))) -> c(a(b(x1))) 8.85/3.08 c(a(b(x1))) -> a(c(b(x1))) 8.85/3.08 Rules from S: 8.85/3.08 none 8.85/3.08 8.85/3.08 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (8) 8.85/3.08 Obligation: 8.85/3.08 Relative term rewrite system: 8.85/3.08 R is empty. 8.85/3.08 The relative TRS consists of the following S rules: 8.85/3.08 8.85/3.08 c(c(a(x1))) -> b(c(c(x1))) 8.85/3.08 8.85/3.08 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (9) RIsEmptyProof (EQUIVALENT) 8.85/3.08 The TRS R is empty. Hence, termination is trivially proven. 8.85/3.08 ---------------------------------------- 8.85/3.08 8.85/3.08 (10) 8.85/3.08 YES 9.00/3.12 EOF