Prove2Me

Bounds for the derivatives of the core clock factor

Proved
BianchiIX.core_small_clock_derivatives

by trppppp · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

Suppose r≥5r\geq5, 0≤ε≤1/1000\leq\varepsilon\leq1/100, 0≤u,w≤ε0\leq u,w\leq\varepsilon, and v>0v>0. Then

∣2(u−w)(u−2w)r3∣≤εr2,∣−2(u−w)r2∣≤2εr2,\left|\frac{2(u-w)(u-2w)}{r^3}\right|\leq\frac\varepsilon{r^2}, \qquad \left|\frac{-2(u-w)}{r^2}\right|\leq\frac{2\varepsilon}{r^2}, ∣2(u−w)vr2∣≤2εvr2.\left|\frac{2(u-w)}{vr^2}\right|\leq\frac{2\varepsilon}{vr^2}.
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_small_clock_derivatives {r m u w v : ℝ} (hr : 5 ≤ r)
    (hm : 0 ≤ m ∧ m ≤ 1 / 100) (hu : 0 ≤ u ∧ u ≤ m) (hw : 0 ≤ w ∧ w ≤ m)
    (hv : 0 < v) :
    |2 * (u - w) * (u - 2 * w) / r ^ 3| ≤ m / r ^ 2 ∧
    |-2 * (u - w) / r ^ 2| ≤ 2 * m / r ^ 2 ∧
    |2 * (u - w) / (v * r ^ 2)| ≤ 2 * m / (v * r ^ 2) := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Algebra/CoreCoefficientEstimates.lean#L93-L118

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