Prove2Me

Three denominator corrections in the weak logarithmic rate

Proved
BianchiIX.core_log_weak_products

by trppppp · Nov 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

Suppose r≥5r\geq5, v≥1v\geq1, and 0≤μ,c≤1/1000\leq\mu,c\leq1/100. Then

2(1+c)rμr2≤1r2,2(1+c)r2μr2≤1,2(1+c)r2μvr2≤μr.\frac{2(1+c)}r\frac\mu{r^2}\leq\frac1{r^2},\qquad \frac{2(1+c)}r\frac{2\mu}{r^2}\leq1,\qquad \frac{2(1+c)}r\frac{2\mu}{vr^2}\leq\frac\mu r.
Preamble
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.FieldSimp

namespace BianchiIX
end BianchiIX

noncomputable section
open BianchiIX
Formal statement
theorem BianchiIX.core_log_weak_products {r m v c : ℝ} (hr : 5 ≤ r)
    (hm : 0 ≤ m ∧ m ≤ 1 / 100) (hv : 1 ≤ v) (hc : 0 ≤ c ∧ c ≤ 1 / 100) :
    (2 * (1 + c) / r) * (m / r ^ 2) ≤ 1 / r ^ 2 ∧
      (2 * (1 + c) / r) * (2 * m / r ^ 2) ≤ 1 ∧
      (2 * (1 + c) / r) * (2 * m / (v * r ^ 2)) ≤ m / r := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Algebra/CoreLogRateEstimates.lean#L21-L45

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