/- SamplingOptimisationGeometryBench public-dev reference.
   Reuses the existing project proof; retain LICENSE and NOTICE. -/
import AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient
import Mathlib.Analysis.InnerProductSpace.Rayleigh

noncomputable section
open Metric InnerProductSpace
open scoped Manifold InnerProductSpace
open AutoSamplingTheory.TechnicalLemmas.Geometry
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]

example
    (n : ℕ) [Fact (Module.finrank ℝ E = n + 1)]
    (A : E →L[ℝ] E) (hA : A.IsSymmetric) (x : sphere (0 : E) 1) :
    ∀ z : (ℝ ∙ (x : E))ᗮ,
      ((∀ w : TangentSpace (𝓡 n) x,
        mvfderiv (𝓡 n) (fun y : sphere (0 : E) 1 => ⟪(y : E), A (y : E)⟫_ℝ) x w =
          ⟪mvfderiv (𝓡 n) ((↑) : sphere (0 : E) 1 → E) x w, (z : E)⟫_ℝ) ↔
        (z : E) = (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E))) := by
  -- Replace this comment with your proof.