Twice continuous differentiability of a moving interval integral

Proved

BianchiIX.contDiffAt_moving_intervalIntegral_two

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

bianchi-ix-era-statisticsdynamical-systems

Let $E$ be a real normed space, $F:E\times\mathbb R\to\mathbb R$ , $\tau:E\to\mathbb R$ , and $x\in E$ . Assume $\tau(x)\geq0$ and define

M_F(y)=\int_0^{\tau(y)}F(y,t)\,dt,\qquad R(y,s)=\tau(y)F(y,s\tau(y)).

All interval integrals are oriented integrals. If $\tau$ is $C^2$ at $x$ and $F$ is $C^2$ at every $(x,t)$ with $t\in[0,\tau(x)]$ , then $M_F$ is $C^2$ at $x$ .

Preamble

import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Topology.Order.Compact
import Mathlib.MeasureTheory.Integral.DominatedConvergence

namespace BianchiIX
end BianchiIX

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

Formal statement

theorem BianchiIX.contDiffAt_moving_intervalIntegral_two
    {F : E × ℝ → ℝ} {τ : E → ℝ} {x : E} (hτ : ContDiffAt ℝ 2 τ x) (hpos : 0 ≤ τ x)
    (hF : ∀ t ∈ Icc 0 (τ x), ContDiffAt ℝ 2 F (x, t)) :
    ContDiffAt ℝ 2 (fun y => ∫ t in 0..τ y, F (y, t)) x := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Analysis/CompactIntegralSecondRegularity.lean#L30-L46

View graph