9.04/3.11 YES 9.80/3.29 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 9.80/3.29 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.80/3.29 9.80/3.29 9.80/3.29 Termination of the given RelTRS could be proven: 9.80/3.29 9.80/3.29 (0) RelTRS 9.80/3.29 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 9.80/3.29 (2) RelTRS 9.80/3.29 (3) RelTRSRRRProof [EQUIVALENT, 93 ms] 9.80/3.29 (4) RelTRS 9.80/3.29 (5) RelTRSRRRProof [EQUIVALENT, 12 ms] 9.80/3.29 (6) RelTRS 9.80/3.29 (7) RelTRSRRRProof [EQUIVALENT, 2 ms] 9.80/3.29 (8) RelTRS 9.80/3.29 (9) RIsEmptyProof [EQUIVALENT, 0 ms] 9.80/3.29 (10) YES 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (0) 9.80/3.29 Obligation: 9.80/3.29 Relative term rewrite system: 9.80/3.29 The relative TRS consists of the following R rules: 9.80/3.29 9.80/3.29 b(c(a(x1))) -> d(d(x1)) 9.80/3.29 b(x1) -> c(c(x1)) 9.80/3.29 a(a(x1)) -> a(c(b(a(x1)))) 9.80/3.29 9.80/3.29 The relative TRS consists of the following S rules: 9.80/3.29 9.80/3.29 a(b(x1)) -> d(x1) 9.80/3.29 d(x1) -> a(b(x1)) 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (1) RelTRS Reverse (EQUIVALENT) 9.80/3.29 We have reversed the following relative TRS [REVERSE]: 9.80/3.29 The set of rules R is 9.80/3.29 b(c(a(x1))) -> d(d(x1)) 9.80/3.29 b(x1) -> c(c(x1)) 9.80/3.29 a(a(x1)) -> a(c(b(a(x1)))) 9.80/3.29 9.80/3.29 The set of rules S is 9.80/3.29 a(b(x1)) -> d(x1) 9.80/3.29 d(x1) -> a(b(x1)) 9.80/3.29 9.80/3.29 We have obtained the following relative TRS: 9.80/3.29 The set of rules R is 9.80/3.29 a(c(b(x1))) -> d(d(x1)) 9.80/3.29 b(x1) -> c(c(x1)) 9.80/3.29 a(a(x1)) -> a(b(c(a(x1)))) 9.80/3.29 9.80/3.29 The set of rules S is 9.80/3.29 b(a(x1)) -> d(x1) 9.80/3.29 d(x1) -> b(a(x1)) 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (2) 9.80/3.29 Obligation: 9.80/3.29 Relative term rewrite system: 9.80/3.29 The relative TRS consists of the following R rules: 9.80/3.29 9.80/3.29 a(c(b(x1))) -> d(d(x1)) 9.80/3.29 b(x1) -> c(c(x1)) 9.80/3.29 a(a(x1)) -> a(b(c(a(x1)))) 9.80/3.29 9.80/3.29 The relative TRS consists of the following S rules: 9.80/3.29 9.80/3.29 b(a(x1)) -> d(x1) 9.80/3.29 d(x1) -> b(a(x1)) 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (3) RelTRSRRRProof (EQUIVALENT) 9.80/3.29 We used the following monotonic ordering for rule removal: 9.80/3.29 Matrix interpretation [MATRO] to (N^3, +, *, >=, >) : 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(a(x_1)) = [[0], [0], [0]] + [[1, 0, 2], [0, 0, 1], [0, 0, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(c(x_1)) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 2, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(b(x_1)) = [[0], [2], [0]] + [[1, 0, 0], [0, 0, 2], [0, 0, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(d(x_1)) = [[0], [2], [0]] + [[1, 0, 2], [0, 0, 0], [0, 0, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 9.80/3.29 Rules from R: 9.80/3.29 9.80/3.29 a(c(b(x1))) -> d(d(x1)) 9.80/3.29 Rules from S: 9.80/3.29 none 9.80/3.29 9.80/3.29 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (4) 9.80/3.29 Obligation: 9.80/3.29 Relative term rewrite system: 9.80/3.29 The relative TRS consists of the following R rules: 9.80/3.29 9.80/3.29 b(x1) -> c(c(x1)) 9.80/3.29 a(a(x1)) -> a(b(c(a(x1)))) 9.80/3.29 9.80/3.29 The relative TRS consists of the following S rules: 9.80/3.29 9.80/3.29 b(a(x1)) -> d(x1) 9.80/3.29 d(x1) -> b(a(x1)) 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (5) RelTRSRRRProof (EQUIVALENT) 9.80/3.29 We used the following monotonic ordering for rule removal: 9.80/3.29 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(b(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(c(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(a(x_1)) = [[0], [2]] + [[1, 1], [0, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 <<< 9.80/3.29 POL(d(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 9.80/3.29 >>> 9.80/3.29 9.80/3.29 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 9.80/3.29 Rules from R: 9.80/3.29 9.80/3.29 a(a(x1)) -> a(b(c(a(x1)))) 9.80/3.29 Rules from S: 9.80/3.29 none 9.80/3.29 9.80/3.29 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (6) 9.80/3.29 Obligation: 9.80/3.29 Relative term rewrite system: 9.80/3.29 The relative TRS consists of the following R rules: 9.80/3.29 9.80/3.29 b(x1) -> c(c(x1)) 9.80/3.29 9.80/3.29 The relative TRS consists of the following S rules: 9.80/3.29 9.80/3.29 b(a(x1)) -> d(x1) 9.80/3.29 d(x1) -> b(a(x1)) 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (7) RelTRSRRRProof (EQUIVALENT) 9.80/3.29 We used the following monotonic ordering for rule removal: 9.80/3.29 Polynomial interpretation [POLO]: 9.80/3.29 9.80/3.29 POL(a(x_1)) = x_1 9.80/3.29 POL(b(x_1)) = 1 + x_1 9.80/3.29 POL(c(x_1)) = x_1 9.80/3.29 POL(d(x_1)) = 1 + x_1 9.80/3.29 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 9.80/3.29 Rules from R: 9.80/3.29 9.80/3.29 b(x1) -> c(c(x1)) 9.80/3.29 Rules from S: 9.80/3.29 none 9.80/3.29 9.80/3.29 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (8) 9.80/3.29 Obligation: 9.80/3.29 Relative term rewrite system: 9.80/3.29 R is empty. 9.80/3.29 The relative TRS consists of the following S rules: 9.80/3.29 9.80/3.29 b(a(x1)) -> d(x1) 9.80/3.29 d(x1) -> b(a(x1)) 9.80/3.29 9.80/3.29 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (9) RIsEmptyProof (EQUIVALENT) 9.80/3.29 The TRS R is empty. Hence, termination is trivially proven. 9.80/3.29 ---------------------------------------- 9.80/3.29 9.80/3.29 (10) 9.80/3.29 YES 9.86/3.32 EOF