AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementGradientDerivativeSymmetry

import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Positive

/-!
# Symmetry of derivatives of gradient fields

For Chewi's displacement interpolation, positivity of the endpoint derivative
must ultimately come from genuine gradient/Hessian structure rather than a
coordinate-level PSD assumption.

This module isolates the symmetry half of that statement.  If a scalar
potential `phi` has Frechet derivative

`D phi(y) = innerSL R (T y)`

at every point and the vector field `T` is Frechet differentiable at `x` with
derivative `A`, then `A` is symmetric.

The proof sends the derivative of `T` through the continuous-linear Riesz map
`innerSL R`, obtaining the second Frechet derivative of `phi`, and then invokes
Mathlib's `second_derivative_symmetric`.  No convexity or monotonicity is needed
for this symmetry leaf.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas
namespace Measure
namespace DisplacementGradientDerivativeSymmetry

open scoped RealInnerProductSpace

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

/-- The derivative of a globally represented gradient field is symmetric at
every point where that vector field is Frechet differentiable. -/
theorem isSymmetric_fderiv_of_gradient_field
    {phi : E → ℝ} {T : E → E} {A : E →L[ℝ] E} {x : E}
    (hphi : ∀ y, HasFDerivAt phi (innerSL ℝ (T y)) y)
    (hT : HasFDerivAt T A x) :
    A.IsSymmetric := by
  have hdual :
      HasFDerivAt (fun y => innerSL ℝ (T y)) ((innerSL ℝ).comp A) x := by
    simpa [Function.comp_def] using
      (innerSL ℝ).hasFDerivAt.comp x hT
  intro v w
  have hs := second_derivative_symmetric
    (f := phi)
    (f' := fun y => innerSL ℝ (T y))
    (f'' := (innerSL ℝ).comp A)
    hphi hdual v w
  change ⟪A v, w⟫ = ⟪v, A w⟫
  calc
    ⟪A v, w⟫ = ⟪A w, v⟫ := by simpa using hs
    _ = ⟪v, A w⟫ := real_inner_comm _ _

end DisplacementGradientDerivativeSymmetry
end Measure
end TechnicalLemmas
end AutoSamplingTheory