`BookProof.ChapterRadialLaplacian.fderiv_comp_normSq` {G : ℝ → ℝ} {x : E} (hG : ContDiffAt ℝ 2 G (‖x‖ ^ 2)) : (fderiv ℝ fun z : E => G (‖z‖ ^ 2)) =ᶠ[nhds x] fun y => (2 * deriv G (

Proved

BookProof.ChapterRadialLaplacian.fderiv_comp_normSq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterRadialLaplacian.

BookProof.ChapterRadialLaplacian.fderiv_comp_normSq {G : ℝ → ℝ} {x : E} (hG : ContDiffAt ℝ 2 G (‖x‖ ^ 2)) : (fderiv ℝ fun z : E => G (‖z‖ ^ 2)) =ᶠ[nhds x] fun y => (2 * deriv G (‖y‖ ^ 2)) • (innerCLM E y)

Formalization note: Lean 4 identifier BookProof.ChapterRadialLaplacian.fderiv_comp_normSq.

Preamble

-- Generated from ChapterRadialLaplacian.lean — theorem BookProof.ChapterRadialLaplacian.fderiv_comp_normSq
import Mathlib
import Definitions.Def_ChapterRadialLaplacian
open BookProof.ChapterRadialLaplacian



open Filter Laplacian InnerProductSpace
open scoped InnerProductSpace RealInnerProductSpace

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]

Formal statement

theorem BookProof.ChapterRadialLaplacian.fderiv_comp_normSq {G : ℝ → ℝ} {x : E} (hG : ContDiffAt ℝ 2 G (‖x‖ ^ 2)) :
    (fderiv ℝ fun z : E => G (‖z‖ ^ 2)) =ᶠ[nhds x]
      fun y => (2 * deriv G (‖y‖ ^ 2)) • (innerCLM E y) := by sorry

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