openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Numerical /ComplexLogThreeCertificate.lean
| import Mathlib.Analysis.SpecialFunctions.Log.Basic | |
| import Mathlib.Tactic.NormNum | |
| namespace OAI | |
| /-! Exact rational intervals, logarithm bounds and arithmetic circuit soundness. -/ | |
| namespace MatrixMultiplication.Foundation | |
| theorem log_three_upper : Real.log 3 < (1098612289 : ℝ) / 10 ^ 9 := by | |
| apply (Real.log_lt_iff_lt_exp (by norm_num : (0 : ℝ) < 3)).2 | |
| refine lt_of_lt_of_le ?_ (Real.sum_le_exp_of_nonneg (by norm_num) 16) | |
| norm_num [Finset.sum_range_succ, Nat.factorial] | |
| end MatrixMultiplication.Foundation | |
| end OAI | |