lean/OAI/Probability/LogConcave/Sampling/JointInterpolationLaw.lean

import Mathlib
import OAI.Probability.LogConcave.Analysis.InterpolationPotential

namespace OAI

section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology

section UpperProof
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology

namespace LogConcaveSampling
open MeasureTheory
open scoped RealInnerProductSpace

lemma interpolation_conditional_exponent {d : ℕ} (F : Point d → ℝ) (x : Point d)
    (r ρ : ℝ) (y z : Point d) (ha : 1-ρ^2≠0) :
    -primitivePotential F x r z-‖y-ρ • z‖^2/(2*(1-ρ^2))=
      -‖y‖^2/2-conditionalPotential F x r ρ y z := by
  simp only [primitivePotential,conditionalPotential,norm_sub_sq_real,norm_smul,
    Real.norm_eq_abs,mul_pow,sq_abs,inner_smul_right]
  rw [real_inner_comm z y]
  field_simp
  ring

lemma weighted_integral_zero_of_gibbs {d : ℕ} {E : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Point d → ℝ}
    (hc : Continuous H) (hi : Integrable (fun z => Real.exp (-H z)))
    {u : Point d → E} (hu : (∫ z,u z ∂gibbs H)=0) :
    (∫ z,Real.exp (-H z) • u z)=0 := by
  rw [integral_gibbs_vector hc hi] at hu
  exact (smul_eq_zero.mp hu).resolve_left (inv_ne_zero (integral_exp_pos hi).ne')

def jointInterpolationLaw {d : ℕ} (F : Point d → ℝ) (x : Point d) (r ρ : ℝ) :=
  (((gibbs (primitivePotential F x r)).prod (stdGaussian (Point d))).map
    (fun p => (p.1,ρ • p.1+Real.sqrt (1-ρ^2) • p.2)))

theorem interpolation_joint_integral_zero {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
    (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r)
    (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1)
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
    {u : Point d × Point d → E}
    (hi : Integrable u (jointInterpolationLaw F x r ρ))
    (hu : ∀ y,(∫ z,u (z,y) ∂gibbs (conditionalPotential F x r ρ y))=0) :
    (∫ p,u p ∂jointInterpolationLaw F x r ρ)=0 := by
  unfold jointInterpolationLaw at hi ⊢
  have ha := (probability_time hρ0 hρ1).1
  have hσ := (Real.sqrt_pos.mpr ha).ne'
  have hp0 := (partition_pos_of_continuous (hF.continuous_potential x r)).ne'
  have hpt := partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr (by linarith))
  let c := (partition (primitivePotential F x r))⁻¹*
    EulerDensity.noiseCoefficient d (Real.sqrt (1-ρ^2))
  have hc0 : c≠0 := mul_ne_zero (ENNReal.inv_ne_zero.mpr hpt)
    (EulerDensity.noiseCoefficient_positive d hσ).ne'
  have hct : c≠⊤ := ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hp0)
    (EulerDensity.noiseCoefficient_finite _ _)
  let w := fun p : Point d × Point d => Real.exp (-primitivePotential F x r p.1-
    ‖p.2-ρ • p.1‖^2/(2*(1-ρ^2)))
  have hw0 : ∀ p,0≤w p := fun _ => (Real.exp_pos _).le
  have hw : Measurable w := by
    have := (hF.continuous_potential x r).measurable
    dsimp [w]; fun_prop
  have hden : Measurable (fun p => ENNReal.ofReal (w p)) := ENNReal.measurable_ofReal.comp hw
  have hd := EulerDensity.joint_density (hF.continuous_potential x r).measurable
    (fun z => ρ • z) (by fun_prop) hσ
  simp only [Real.sq_sqrt ha.le] at hd
  change _=c • ((volume : Measure (Point d)).prod volume).withDensity
    (fun p => ENNReal.ofReal (w p)) at hd
  rw [hd] at hi ⊢
  have hif : Integrable (fun p => w p • u p) ((volume : Measure (Point d)).prod volume) := by
    have hi' := (integrable_smul_measure hc0 hct).mp hi
    rw [integrable_withDensity_iff_integrable_smul' (hden)
      (Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] at hi'
    simpa only [ENNReal.toReal_ofReal (hw0 _)] using hi'
  rw [integral_smul_measure,integral_withDensity_eq_integral_toReal_smul
    hden (Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
  simp only [ENNReal.toReal_ofReal (hw0 _)]
  rw [integral_prod_symm _ hif]
  have hz : ∀ y,(∫ z,w (z,y) • u (z,y))=0 := by
    intro y
    have hci := (conditional_hasGaussianLowerTail hF x hr hl hρ0 hρ1 y).integrable_exp
      (conditionalPotential_smooth hF x r ρ y).continuous
    have hh := weighted_integral_zero_of_gibbs
      (conditionalPotential_smooth hF x r ρ y).continuous hci (hu y)
    have he : ∀ z,w (z,y)=Real.exp (-‖y‖^2/2)*Real.exp (-conditionalPotential F x r ρ y z) := by
      intro z; dsimp [w]
      rw [interpolation_conditional_exponent F x r ρ y z ha.ne',sub_eq_add_neg,Real.exp_add]
    simp_rw [he,mul_smul]
    rw [integral_smul,hh,smul_zero]
  simp_rw [hz]
  simp
end LogConcaveSampling

end UpperProof
end
end
end

end OAI