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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | 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
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