6.64/2.50 YES 6.90/2.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 6.90/2.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.90/2.51 6.90/2.51 6.90/2.51 Termination of the given RelTRS could be proven: 6.90/2.51 6.90/2.51 (0) RelTRS 6.90/2.51 (1) RelTRSRRRProof [EQUIVALENT, 176 ms] 6.90/2.51 (2) RelTRS 6.90/2.51 (3) RelTRSRRRProof [EQUIVALENT, 0 ms] 6.90/2.51 (4) RelTRS 6.90/2.51 (5) RIsEmptyProof [EQUIVALENT, 5 ms] 6.90/2.51 (6) YES 6.90/2.51 6.90/2.51 6.90/2.51 ---------------------------------------- 6.90/2.51 6.90/2.51 (0) 6.90/2.51 Obligation: 6.90/2.51 Relative term rewrite system: 6.90/2.51 The relative TRS consists of the following R rules: 6.90/2.51 6.90/2.51 b(b(a(x1))) -> a(a(b(x1))) 6.90/2.51 a(a(a(x1))) -> c(c(a(x1))) 6.90/2.51 a(a(a(x1))) -> b(c(b(x1))) 6.90/2.51 b(a(c(x1))) -> a(b(a(x1))) 6.90/2.51 6.90/2.51 The relative TRS consists of the following S rules: 6.90/2.51 6.90/2.51 c(c(b(x1))) -> a(c(b(x1))) 6.90/2.51 a(a(a(x1))) -> b(a(a(x1))) 6.90/2.51 6.90/2.51 6.90/2.51 ---------------------------------------- 6.90/2.51 6.90/2.51 (1) RelTRSRRRProof (EQUIVALENT) 6.90/2.51 We used the following monotonic ordering for rule removal: 6.90/2.51 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 6.90/2.51 6.90/2.51 <<< 6.90/2.51 POL(b(x_1)) = [[0], [0]] + [[2, 1], [0, 1]] * x_1 6.90/2.51 >>> 6.90/2.51 6.90/2.51 <<< 6.90/2.51 POL(a(x_1)) = [[0], [1]] + [[2, 2], [0, 0]] * x_1 6.90/2.51 >>> 6.90/2.51 6.90/2.51 <<< 6.90/2.51 POL(c(x_1)) = [[0], [1]] + [[2, 2], [0, 0]] * x_1 6.90/2.51 >>> 6.90/2.51 6.90/2.51 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 6.90/2.51 Rules from R: 6.90/2.51 6.90/2.51 b(b(a(x1))) -> a(a(b(x1))) 6.90/2.51 a(a(a(x1))) -> b(c(b(x1))) 6.90/2.51 b(a(c(x1))) -> a(b(a(x1))) 6.90/2.51 Rules from S: 6.90/2.51 6.90/2.51 a(a(a(x1))) -> b(a(a(x1))) 6.90/2.51 6.90/2.51 6.90/2.51 6.90/2.51 6.90/2.51 ---------------------------------------- 6.90/2.51 6.90/2.51 (2) 6.90/2.51 Obligation: 6.90/2.51 Relative term rewrite system: 6.90/2.51 The relative TRS consists of the following R rules: 6.90/2.51 6.90/2.51 a(a(a(x1))) -> c(c(a(x1))) 6.90/2.51 6.90/2.51 The relative TRS consists of the following S rules: 6.90/2.51 6.90/2.51 c(c(b(x1))) -> a(c(b(x1))) 6.90/2.51 6.90/2.51 6.90/2.51 ---------------------------------------- 6.90/2.51 6.90/2.51 (3) RelTRSRRRProof (EQUIVALENT) 6.90/2.51 We used the following monotonic ordering for rule removal: 6.90/2.51 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 6.90/2.51 6.90/2.51 <<< 6.90/2.51 POL(a(x_1)) = [[1], [1]] + [[2, 0], [2, 0]] * x_1 6.90/2.51 >>> 6.90/2.51 6.90/2.51 <<< 6.90/2.51 POL(c(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 6.90/2.51 >>> 6.90/2.51 6.90/2.51 <<< 6.90/2.51 POL(b(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 6.90/2.51 >>> 6.90/2.51 6.90/2.51 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 6.90/2.51 Rules from R: 6.90/2.51 6.90/2.51 a(a(a(x1))) -> c(c(a(x1))) 6.90/2.51 Rules from S: 6.90/2.51 none 6.90/2.51 6.90/2.51 6.90/2.51 6.90/2.51 6.90/2.51 ---------------------------------------- 6.90/2.51 6.90/2.51 (4) 6.90/2.51 Obligation: 6.90/2.51 Relative term rewrite system: 6.90/2.51 R is empty. 6.90/2.51 The relative TRS consists of the following S rules: 6.90/2.51 6.90/2.51 c(c(b(x1))) -> a(c(b(x1))) 6.90/2.51 6.90/2.51 6.90/2.51 ---------------------------------------- 6.90/2.51 6.90/2.51 (5) RIsEmptyProof (EQUIVALENT) 6.90/2.51 The TRS R is empty. Hence, termination is trivially proven. 6.90/2.51 ---------------------------------------- 6.90/2.51 6.90/2.51 (6) 6.90/2.51 YES 7.13/2.63 EOF