`BookProof.WeylCauchyRiemann.mollify_analyticOn` {f : ℂ → ℂ} {U : Set ℂ} (hf : LocallyIntegrableOn f U) (hCR : WeakCauchyRiemannOn f U) {c : ℂ} {R r δ : ℝ} (hδ : 0 < δ) (hr : 0 < r

Proved

BookProof.WeylCauchyRiemann.mollify_analyticOn

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWeylCauchyRiemann.

BookProof.WeylCauchyRiemann.mollify_analyticOn {f : ℂ → ℂ} {U : Set ℂ} (hf : LocallyIntegrableOn f U) (hCR : WeakCauchyRiemannOn f U) {c : ℂ} {R r δ : ℝ} (hδ : 0 < δ) (hr : 0 < r) (hrR : r + δ ≤ R) (hKU : closedBall c R ⊆ U) {χ : ℂ → ℝ} (hχ : ContDiff ℝ ∞ χ) (hχc : HasCompactSupport χ) (hsupp : Function.support χ ⊆ ball (0 : ℂ) δ) : AnalyticOn ℂ (mollify ((closedBall c R).indicator f) χ) (ball c r)

Formalization note: Lean 4 identifier BookProof.WeylCauchyRiemann.mollify_analyticOn.

Preamble

-- Generated from ChapterWeylCauchyRiemann.lean — theorem BookProof.WeylCauchyRiemann.mollify_analyticOn
import Definitions.Def_ChapterRadialMollifier
import Mathlib
import Definitions.Def_ChapterWeylCauchyRiemann
open BookProof.WeylCauchyRiemann



open MeasureTheory Metric Set Filter Complex BookProof.RadialMollifier
open scoped Convolution Topology ContDiff

noncomputable section

Formal statement

theorem BookProof.WeylCauchyRiemann.mollify_analyticOn {f : ℂ → ℂ} {U : Set ℂ}
    (hf : LocallyIntegrableOn f U) (hCR : WeakCauchyRiemannOn f U)
    {c : ℂ} {R r δ : ℝ} (hδ : 0 < δ) (hr : 0 < r) (hrR : r + δ ≤ R)
    (hKU : closedBall c R ⊆ U) {χ : ℂ → ℝ} (hχ : ContDiff ℝ ∞ χ)
    (hχc : HasCompactSupport χ) (hsupp : Function.support χ ⊆ ball (0 : ℂ) δ) :
    AnalyticOn ℂ (mollify ((closedBall c R).indicator f) χ) (ball c r) := by sorry

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