`BookProof.WeylCauchyRiemann.dbar_ofReal_comp_sub` {χ : ℂ → ℝ} (hχ : ContDiff ℝ ∞ χ) (z w : ℂ) : dbar (fun v => ((χ (z - v) : ℝ) : ℂ)) w = -dbarR χ (z - w)

Proved

BookProof.WeylCauchyRiemann.dbar_ofReal_comp_sub

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWeylCauchyRiemann.

BookProof.WeylCauchyRiemann.dbar_ofReal_comp_sub {χ : ℂ → ℝ} (hχ : ContDiff ℝ ∞ χ) (z w : ℂ) : dbar (fun v => ((χ (z - v) : ℝ) : ℂ)) w = -dbarR χ (z - w)

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

Preamble

-- Generated from ChapterWeylCauchyRiemann.lean — theorem BookProof.WeylCauchyRiemann.dbar_ofReal_comp_sub
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.dbar_ofReal_comp_sub {χ : ℂ → ℝ} (hχ : ContDiff ℝ ∞ χ) (z w : ℂ) :
    dbar (fun v => ((χ (z - v) : ℝ) : ℂ)) w = -dbarR χ (z - w) := by sorry

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