7.03/2.59 YES 7.03/2.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 7.03/2.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.03/2.60 7.03/2.60 7.03/2.60 Termination of the given RelTRS could be proven: 7.03/2.60 7.03/2.60 (0) RelTRS 7.03/2.60 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 7.03/2.60 (2) RelTRS 7.03/2.60 (3) RelTRSRRRProof [EQUIVALENT, 42 ms] 7.03/2.60 (4) RelTRS 7.03/2.60 (5) RelTRSRRRProof [EQUIVALENT, 19 ms] 7.03/2.60 (6) RelTRS 7.03/2.60 (7) RelTRSRRRProof [EQUIVALENT, 4 ms] 7.03/2.60 (8) RelTRS 7.03/2.60 (9) RIsEmptyProof [EQUIVALENT, 1 ms] 7.03/2.60 (10) YES 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (0) 7.03/2.60 Obligation: 7.03/2.60 Relative term rewrite system: 7.03/2.60 The relative TRS consists of the following R rules: 7.03/2.60 7.03/2.60 b(b(x1)) -> c(d(x1)) 7.03/2.60 c(c(x1)) -> d(d(d(x1))) 7.03/2.60 d(d(d(x1))) -> a(c(x1)) 7.03/2.60 7.03/2.60 The relative TRS consists of the following S rules: 7.03/2.60 7.03/2.60 a(a(x1)) -> b(c(x1)) 7.03/2.60 b(c(x1)) -> a(a(x1)) 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (1) RelTRS Reverse (EQUIVALENT) 7.03/2.60 We have reversed the following relative TRS [REVERSE]: 7.03/2.60 The set of rules R is 7.03/2.60 b(b(x1)) -> c(d(x1)) 7.03/2.60 c(c(x1)) -> d(d(d(x1))) 7.03/2.60 d(d(d(x1))) -> a(c(x1)) 7.03/2.60 7.03/2.60 The set of rules S is 7.03/2.60 a(a(x1)) -> b(c(x1)) 7.03/2.60 b(c(x1)) -> a(a(x1)) 7.03/2.60 7.03/2.60 We have obtained the following relative TRS: 7.03/2.60 The set of rules R is 7.03/2.60 b(b(x1)) -> d(c(x1)) 7.03/2.60 c(c(x1)) -> d(d(d(x1))) 7.03/2.60 d(d(d(x1))) -> c(a(x1)) 7.03/2.60 7.03/2.60 The set of rules S is 7.03/2.60 a(a(x1)) -> c(b(x1)) 7.03/2.60 c(b(x1)) -> a(a(x1)) 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (2) 7.03/2.60 Obligation: 7.03/2.60 Relative term rewrite system: 7.03/2.60 The relative TRS consists of the following R rules: 7.03/2.60 7.03/2.60 b(b(x1)) -> d(c(x1)) 7.03/2.60 c(c(x1)) -> d(d(d(x1))) 7.03/2.60 d(d(d(x1))) -> c(a(x1)) 7.03/2.60 7.03/2.60 The relative TRS consists of the following S rules: 7.03/2.60 7.03/2.60 a(a(x1)) -> c(b(x1)) 7.03/2.60 c(b(x1)) -> a(a(x1)) 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (3) RelTRSRRRProof (EQUIVALENT) 7.03/2.60 We used the following monotonic ordering for rule removal: 7.03/2.60 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(b(x_1)) = [[1], [2]] + [[1, 1], [0, 0]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(d(x_1)) = [[0], [2]] + [[1, 1], [0, 0]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(c(x_1)) = [[1], [2]] + [[1, 1], [0, 0]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(a(x_1)) = [[1], [2]] + [[1, 1], [0, 0]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.03/2.60 Rules from R: 7.03/2.60 7.03/2.60 b(b(x1)) -> d(c(x1)) 7.03/2.60 Rules from S: 7.03/2.60 none 7.03/2.60 7.03/2.60 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (4) 7.03/2.60 Obligation: 7.03/2.60 Relative term rewrite system: 7.03/2.60 The relative TRS consists of the following R rules: 7.03/2.60 7.03/2.60 c(c(x1)) -> d(d(d(x1))) 7.03/2.60 d(d(d(x1))) -> c(a(x1)) 7.03/2.60 7.03/2.60 The relative TRS consists of the following S rules: 7.03/2.60 7.03/2.60 a(a(x1)) -> c(b(x1)) 7.03/2.60 c(b(x1)) -> a(a(x1)) 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (5) RelTRSRRRProof (EQUIVALENT) 7.03/2.60 We used the following monotonic ordering for rule removal: 7.03/2.60 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(c(x_1)) = [[1], [1]] + [[1, 1], [0, 0]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(d(x_1)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(a(x_1)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 <<< 7.03/2.60 POL(b(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 7.03/2.60 >>> 7.03/2.60 7.03/2.60 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.03/2.60 Rules from R: 7.03/2.60 7.03/2.60 c(c(x1)) -> d(d(d(x1))) 7.03/2.60 Rules from S: 7.03/2.60 none 7.03/2.60 7.03/2.60 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (6) 7.03/2.60 Obligation: 7.03/2.60 Relative term rewrite system: 7.03/2.60 The relative TRS consists of the following R rules: 7.03/2.60 7.03/2.60 d(d(d(x1))) -> c(a(x1)) 7.03/2.60 7.03/2.60 The relative TRS consists of the following S rules: 7.03/2.60 7.03/2.60 a(a(x1)) -> c(b(x1)) 7.03/2.60 c(b(x1)) -> a(a(x1)) 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (7) RelTRSRRRProof (EQUIVALENT) 7.03/2.60 We used the following monotonic ordering for rule removal: 7.03/2.60 Polynomial interpretation [POLO]: 7.03/2.60 7.03/2.60 POL(a(x_1)) = 1 + x_1 7.03/2.60 POL(b(x_1)) = 1 + x_1 7.03/2.60 POL(c(x_1)) = 1 + x_1 7.03/2.60 POL(d(x_1)) = 1 + x_1 7.03/2.60 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 7.03/2.60 Rules from R: 7.03/2.60 7.03/2.60 d(d(d(x1))) -> c(a(x1)) 7.03/2.60 Rules from S: 7.03/2.60 none 7.03/2.60 7.03/2.60 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (8) 7.03/2.60 Obligation: 7.03/2.60 Relative term rewrite system: 7.03/2.60 R is empty. 7.03/2.60 The relative TRS consists of the following S rules: 7.03/2.60 7.03/2.60 a(a(x1)) -> c(b(x1)) 7.03/2.60 c(b(x1)) -> a(a(x1)) 7.03/2.60 7.03/2.60 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (9) RIsEmptyProof (EQUIVALENT) 7.03/2.60 The TRS R is empty. Hence, termination is trivially proven. 7.03/2.60 ---------------------------------------- 7.03/2.60 7.03/2.60 (10) 7.03/2.60 YES 7.03/2.63 EOF