AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexGradientPositive

import AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexPotentialSupport
import AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementGradientDerivativeSymmetry
import AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMonotoneDerivative
import AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator

/-!
# Positive derivatives of differentiable convex gradient fields

This module is a pure composition node in the Chewi 1.4.5 Brenier/Jacobian
spine.  The ingredients are already isolated independently:

* convexity plus the global Riesz representation `D phi = innerSL R ∘ T`
  gives monotonicity of `T`;
* monotonicity plus differentiability of `T` at `x` gives the nonnegative
  quadratic form of the derivative;
* the same gradient representation plus differentiability of `T` at `x`
  gives symmetry of the derivative via Mathlib's second-derivative symmetry
  theorem.

Combining the last two properties yields a positive continuous-linear
operator.  No new regularity or optimal-transport assumption is introduced
here.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas
namespace Measure
namespace DisplacementConvexGradientPositive

open scoped RealInnerProductSpace

noncomputable section

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

/-- At every point where the gradient vector field of a globally convex
potential is Frechet differentiable, its derivative is a positive operator.
This statement is dimension-free. -/
theorem isPositive_fderiv_of_convex_gradient_field
    {phi : E → ℝ} {T : E → E} {A : E →L[ℝ] E} {x : E}
    (hconv : ConvexOn ℝ Set.univ phi)
    (hphi : ∀ y, HasFDerivAt phi (innerSL ℝ (T y)) y)
    (hT : HasFDerivAt T A x) :
    A.IsPositive := by
  have hmono : DisplacementMapInjectivity.IsMonotoneMap T :=
    DisplacementConvexPotentialSupport.isMonotoneMap_of_convexOn_univ_hasFDerivAt_inner
      hconv hphi
  have hsymm : A.IsSymmetric :=
    DisplacementGradientDerivativeSymmetry.isSymmetric_fderiv_of_gradient_field
      hphi hT
  exact
    DisplacementMonotoneDerivative.isPositive_fderiv_of_monotone_of_isSymmetric
      hmono hT hsymm

section FiniteDimensional

variable {ι : Type*}
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [Fintype ι] [DecidableEq ι]

/-- In an orthonormal basis, the same local derivative therefore has a PSD
matrix representation.  The basis is only a coordinate witness; positivity is
proved before choosing it. -/
theorem toMatrix_fderiv_posSemidef_of_convex_gradient_field
    (b : OrthonormalBasis ι ℝ E)
    {phi : E → ℝ} {T : E → E} {A : E →L[ℝ] E} {x : E}
    (hconv : ConvexOn ℝ Set.univ phi)
    (hphi : ∀ y, HasFDerivAt phi (innerSL ℝ (T y)) y)
    (hT : HasFDerivAt T A x) :
    (LinearMap.toMatrix b.toBasis b.toBasis A.toLinearMap).PosSemidef :=
  DisplacementPositiveOperator.toMatrix_posSemidef_of_isPositive
    b A (isPositive_fderiv_of_convex_gradient_field hconv hphi hT)

/-- Consequently, the interior affine displacement derivative has strictly
positive determinant at every such differentiability point. -/
theorem det_affineDisplacementDerivative_pos_of_convex_gradient_field
    (b : OrthonormalBasis ι ℝ E)
    {phi : E → ℝ} {T : E → E} {A : E →L[ℝ] E} {x : E}
    (hconv : ConvexOn ℝ Set.univ phi)
    (hphi : ∀ y, HasFDerivAt phi (innerSL ℝ (T y)) y)
    (hT : HasFDerivAt T A x)
    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t < 1) :
    0 < LinearMap.det
      (DisplacementMapDerivative.affineDisplacementDerivative A t).toLinearMap :=
  DisplacementPositiveOperator.det_affineDisplacementDerivative_pos_of_isPositive
    b A (isPositive_fderiv_of_convex_gradient_field hconv hphi hT) t ht0 ht1

end FiniteDimensional

end

end DisplacementConvexGradientPositive
end Measure
end TechnicalLemmas
end AutoSamplingTheory