3.61/1.76 YES 3.61/1.77 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.61/1.77 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.61/1.77 3.61/1.77 3.61/1.77 Termination of the given RelTRS could be proven: 3.61/1.77 3.61/1.77 (0) RelTRS 3.61/1.77 (1) RelTRSRRRProof [EQUIVALENT, 37 ms] 3.61/1.77 (2) RelTRS 3.61/1.77 (3) RIsEmptyProof [EQUIVALENT, 0 ms] 3.61/1.77 (4) YES 3.61/1.77 3.61/1.77 3.61/1.77 ---------------------------------------- 3.61/1.77 3.61/1.77 (0) 3.61/1.77 Obligation: 3.61/1.77 Relative term rewrite system: 3.61/1.77 The relative TRS consists of the following R rules: 3.61/1.77 3.61/1.77 f(x) -> x 3.61/1.77 3.61/1.77 The relative TRS consists of the following S rules: 3.61/1.77 3.61/1.77 a -> g(a) 3.61/1.77 3.61/1.77 3.61/1.77 ---------------------------------------- 3.61/1.77 3.61/1.77 (1) RelTRSRRRProof (EQUIVALENT) 3.61/1.77 We used the following monotonic ordering for rule removal: 3.61/1.77 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 3.61/1.77 3.61/1.77 <<< 3.61/1.77 POL(f(x_1)) = [[1], [0]] + [[1, 0], [0, 1]] * x_1 3.61/1.77 >>> 3.61/1.77 3.61/1.77 <<< 3.61/1.77 POL(a) = [[0], [0]] 3.61/1.77 >>> 3.61/1.77 3.61/1.77 <<< 3.61/1.77 POL(g(x_1)) = [[0], [0]] + [[2, 1], [2, 2]] * x_1 3.61/1.77 >>> 3.61/1.77 3.61/1.77 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 3.61/1.77 Rules from R: 3.61/1.77 3.61/1.77 f(x) -> x 3.61/1.77 Rules from S: 3.61/1.77 none 3.61/1.77 3.61/1.77 3.61/1.77 3.61/1.77 3.61/1.77 ---------------------------------------- 3.61/1.77 3.61/1.77 (2) 3.61/1.77 Obligation: 3.61/1.77 Relative term rewrite system: 3.61/1.77 R is empty. 3.61/1.77 The relative TRS consists of the following S rules: 3.61/1.77 3.61/1.77 a -> g(a) 3.61/1.77 3.61/1.77 3.61/1.77 ---------------------------------------- 3.61/1.77 3.61/1.77 (3) RIsEmptyProof (EQUIVALENT) 3.61/1.77 The TRS R is empty. Hence, termination is trivially proven. 3.61/1.77 ---------------------------------------- 3.61/1.77 3.61/1.77 (4) 3.61/1.77 YES 3.90/1.80 EOF