`BookProof.ChapterRadialLaplacian.laplacian_comp_normSq` [FiniteDimensional ℝ E] {G : ℝ → ℝ} {x : E} (hG : ContDiffAt ℝ 2 G (‖x‖ ^ 2)) : (Δ fun y : E => G (‖y‖ ^ 2)) x = 4 *...

Proved

BookProof.ChapterRadialLaplacian.laplacian_comp_normSq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterRadialLaplacian.

BookProof.ChapterRadialLaplacian.laplacian_comp_normSq [FiniteDimensional ℝ E] {G : ℝ → ℝ} {x : E} (hG : ContDiffAt ℝ 2 G (‖x‖ ^ 2)) : (Δ fun y : E => G (‖y‖ ^ 2)) x = 4 * ‖x‖ ^ 2 * deriv (deriv G) (‖x‖ ^ 2) + 2 * (Module.finrank ℝ E) * deriv G (‖x‖ ^ 2)

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

Preamble

-- Generated from ChapterRadialLaplacian.lean — theorem BookProof.ChapterRadialLaplacian.laplacian_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.laplacian_comp_normSq [FiniteDimensional ℝ E] {G : ℝ → ℝ} {x : E}
    (hG : ContDiffAt ℝ 2 G (‖x‖ ^ 2)) :
    (Δ fun y : E => G (‖y‖ ^ 2)) x
      = 4 * ‖x‖ ^ 2 * deriv (deriv G) (‖x‖ ^ 2)
        + 2 * (Module.finrank ℝ E) * deriv G (‖x‖ ^ 2) := by sorry

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