The two-direction Gronwall estimate in a real normed space

Proved

BianchiIX.ode_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 $E$ be a real normed space and $f,df:\mathbb R\to E$ . 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 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

namespace BianchiIX
end BianchiIX

noncomputable section
open BianchiIX
open Set Filter Metric Asymptotics
open scoped Topology NNReal
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BianchiIX.ode_norm_le_gronwall_uIcc {f df : ℝ → E} {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/Analysis/ODEVariation.lean#L12-L34

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