55.68/14.89 YES 55.68/14.91 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 55.68/14.91 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 55.68/14.91 55.68/14.91 55.68/14.91 Termination of the given RelTRS could be proven: 55.68/14.91 55.68/14.91 (0) RelTRS 55.68/14.91 (1) RelTRSRRRProof [EQUIVALENT, 946 ms] 55.68/14.91 (2) RelTRS 55.68/14.91 (3) RelTRSRRRProof [EQUIVALENT, 50 ms] 55.68/14.91 (4) RelTRS 55.68/14.91 (5) RelTRSRRRProof [EQUIVALENT, 8 ms] 55.68/14.91 (6) RelTRS 55.68/14.91 (7) RIsEmptyProof [EQUIVALENT, 0 ms] 55.68/14.91 (8) YES 55.68/14.91 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (0) 55.68/14.91 Obligation: 55.68/14.91 Relative term rewrite system: 55.68/14.91 The relative TRS consists of the following R rules: 55.68/14.91 55.68/14.91 b(a(a(x1))) -> c(a(c(x1))) 55.68/14.91 a(c(a(x1))) -> b(c(a(x1))) 55.68/14.91 55.68/14.91 The relative TRS consists of the following S rules: 55.68/14.91 55.68/14.91 b(b(c(x1))) -> c(c(a(x1))) 55.68/14.91 a(b(b(x1))) -> b(c(b(x1))) 55.68/14.91 c(c(b(x1))) -> b(a(b(x1))) 55.68/14.91 b(a(a(x1))) -> a(c(a(x1))) 55.68/14.91 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (1) RelTRSRRRProof (EQUIVALENT) 55.68/14.91 We used the following monotonic ordering for rule removal: 55.68/14.91 Matrix interpretation [MATRO] to (N^3, +, *, >=, >) : 55.68/14.91 55.68/14.91 <<< 55.68/14.91 POL(b(x_1)) = [[0], [0], [0]] + [[2, 0, 1], [2, 0, 1], [2, 0, 1]] * x_1 55.68/14.91 >>> 55.68/14.91 55.68/14.91 <<< 55.68/14.91 POL(a(x_1)) = [[0], [2], [0]] + [[3, 0, 0], [3, 0, 0], [2, 1, 0]] * x_1 55.68/14.91 >>> 55.68/14.91 55.68/14.91 <<< 55.68/14.91 POL(c(x_1)) = [[0], [0], [0]] + [[3, 0, 0], [3, 0, 0], [3, 0, 0]] * x_1 55.68/14.91 >>> 55.68/14.91 55.68/14.91 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 55.68/14.91 Rules from R: 55.68/14.91 55.68/14.91 b(a(a(x1))) -> c(a(c(x1))) 55.68/14.91 Rules from S: 55.68/14.91 55.68/14.91 b(a(a(x1))) -> a(c(a(x1))) 55.68/14.91 55.68/14.91 55.68/14.91 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (2) 55.68/14.91 Obligation: 55.68/14.91 Relative term rewrite system: 55.68/14.91 The relative TRS consists of the following R rules: 55.68/14.91 55.68/14.91 a(c(a(x1))) -> b(c(a(x1))) 55.68/14.91 55.68/14.91 The relative TRS consists of the following S rules: 55.68/14.91 55.68/14.91 b(b(c(x1))) -> c(c(a(x1))) 55.68/14.91 a(b(b(x1))) -> b(c(b(x1))) 55.68/14.91 c(c(b(x1))) -> b(a(b(x1))) 55.68/14.91 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (3) RelTRSRRRProof (EQUIVALENT) 55.68/14.91 We used the following monotonic ordering for rule removal: 55.68/14.91 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 55.68/14.91 55.68/14.91 <<< 55.68/14.91 POL(a(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 55.68/14.91 >>> 55.68/14.91 55.68/14.91 <<< 55.68/14.91 POL(c(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 55.68/14.91 >>> 55.68/14.91 55.68/14.91 <<< 55.68/14.91 POL(b(x_1)) = [[1], [0]] + [[2, 0], [2, 0]] * x_1 55.68/14.91 >>> 55.68/14.91 55.68/14.91 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 55.68/14.91 Rules from R: 55.68/14.91 none 55.68/14.91 Rules from S: 55.68/14.91 55.68/14.91 b(b(c(x1))) -> c(c(a(x1))) 55.68/14.91 a(b(b(x1))) -> b(c(b(x1))) 55.68/14.91 55.68/14.91 55.68/14.91 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (4) 55.68/14.91 Obligation: 55.68/14.91 Relative term rewrite system: 55.68/14.91 The relative TRS consists of the following R rules: 55.68/14.91 55.68/14.91 a(c(a(x1))) -> b(c(a(x1))) 55.68/14.91 55.68/14.91 The relative TRS consists of the following S rules: 55.68/14.91 55.68/14.91 c(c(b(x1))) -> b(a(b(x1))) 55.68/14.91 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (5) RelTRSRRRProof (EQUIVALENT) 55.68/14.91 We used the following monotonic ordering for rule removal: 55.68/14.91 Knuth-Bendix order [KBO] with precedence:c_1 > a_1 > b_1 55.68/14.91 55.68/14.91 and weight map: 55.68/14.91 55.68/14.91 a_1=3 55.68/14.91 c_1=2 55.68/14.91 b_1=1 55.68/14.91 55.68/14.91 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 55.68/14.91 Rules from R: 55.68/14.91 55.68/14.91 a(c(a(x1))) -> b(c(a(x1))) 55.68/14.91 Rules from S: 55.68/14.91 55.68/14.91 c(c(b(x1))) -> b(a(b(x1))) 55.68/14.91 55.68/14.91 55.68/14.91 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (6) 55.68/14.91 Obligation: 55.68/14.91 Relative term rewrite system: 55.68/14.91 R is empty. 55.68/14.91 S is empty. 55.68/14.91 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (7) RIsEmptyProof (EQUIVALENT) 55.68/14.91 The TRS R is empty. Hence, termination is trivially proven. 55.68/14.91 ---------------------------------------- 55.68/14.91 55.68/14.91 (8) 55.68/14.91 YES 55.82/14.94 EOF