A Gronwall norm bound in either time direction

Proved

BianchiIX.norm_le_gronwall_uIcc

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

bianchi-ix-era-statisticsdynamical-systems

For real $\delta,K,\varepsilon,\ell$ , let

G(\delta,K,\varepsilon,\ell)= \begin{cases} \delta+\varepsilon\ell,&K=0,\\ \delta e^{K\ell}+\varepsilon(e^{K\ell}-1)/K,&K\ne0. \end{cases}

Write $[s,t]_{\mathrm{u}}=[\min(s,t),\max(s,t)]$ . Let $f,df:\mathbb R\to\mathbb R^5$ , with the finite-product norm. Suppose $f'(u)=df(u)$ on $[t_0,t]_{\mathrm u}$ , $\|f(t_0)\|\leq\delta$ , and

\|df(u)\|\leq K\|f(u)\|+\varepsilon\quad(u\in[t_0,t]_{\mathrm u}).

Then $\|f(t)\|\leq G(\delta,K,\varepsilon,|t-t_0|)$ .

Preamble

import Definitions.Def_BianchiIX_Dynamics_Normalized
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.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique

noncomputable section
open BianchiIX
open Set Filter Metric
open scoped Topology NNReal

Formal statement

theorem BianchiIX.norm_le_gronwall_uIcc {f df : ℝ → NormalizedState} {t₀ t δ K ε : ℝ}
    (hf : ∀ u ∈ uIcc t₀ t, HasDerivAt f (df u) u)
    (hinit : ‖f t₀‖ ≤ δ)
    (hbound : ∀ u ∈ uIcc t₀ t, ‖df u‖ ≤ K * ‖f u‖ + ε) :
    ‖f t‖ ≤ gronwallBound δ K ε |t - t₀| := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/VariationRemainder.lean#L10-L33

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