import Mathlib
import OAI.Probability.LogConcave.Sampling.CoordinateProjection

namespace OAI

section
section
noncomputable section
namespace LogConcaveSampling
open MeasureTheory Set ProbabilityTheory
open scoped ENNReal NNReal

lemma productPotential_measurable {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
    (hH : Measurable H) (hG : Measurable G) : Measurable (productPotential H G) :=
  (hH.comp ((productPointEquiv d e).continuous.measurable.fst)).add
    (hG.comp ((productPointEquiv d e).continuous.measurable.snd))

lemma productPotential_unnormalized {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
    (hH : Measurable H) (hG : Measurable G) :
    (volume.withDensity (gibbsDensity (productPotential H G))).map (productPointEquiv d e)=
      (volume.withDensity (gibbsDensity H)).prod (volume.withDensity (gibbsDensity G)) := by
  let E := (productPointEquiv d e).toHomeomorph.toMeasurableEquiv
  have hd : Measurable (gibbsDensity (productPotential H G)) :=
    (Real.measurable_exp.comp (productPotential_measurable hH hG).neg).ennreal_ofReal
  have hHd : Measurable (gibbsDensity H) := (Real.measurable_exp.comp hH.neg).ennreal_ofReal
  have hGd : Measurable (gibbsDensity G) := (Real.measurable_exp.comp hG.neg).ennreal_ofReal
  rw [show (productPointEquiv d e : Point (d+e) → Point d × Point e)=E from rfl,
    map_withDensity_of_measurePreserving E (productPointEquiv_volume d e) hd,
    prod_withDensity hHd hGd]
  congr 1
  funext p
  rw [productPotential_density]
  change gibbsDensity H (E (E.symm p)).1*gibbsDensity G (E (E.symm p)).2=_
  rw [E.apply_symm_apply]

lemma productPotential_partition {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
    (hH : Measurable H) (hG : Measurable G) :
    partition (productPotential H G)=partition H*partition G := by
  have he := congrArg (fun μ : Measure (Point d × Point e) => μ univ)
    (productPotential_unnormalized hH hG)
  rw [Measure.map_apply (productPointEquiv d e).continuous.measurable MeasurableSet.univ,
    preimage_univ] at he
  have hp : ((volume.withDensity (gibbsDensity H)).prod (volume.withDensity (gibbsDensity G))) univ=
      (volume.withDensity (gibbsDensity H)) univ*(volume.withDensity (gibbsDensity G)) univ := by
    rw [← univ_prod_univ,Measure.prod_prod]
  rw [hp] at he
  simpa only [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,partition] using he

theorem productPotential_gibbs {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
    (hH : Continuous H) (hG : Continuous G) :
    (gibbs (productPotential H G)).map (productPointEquiv d e)=(gibbs H).prod (gibbs G) := by
  unfold gibbs
  rw [Measure.map_smul _ (productPointEquiv d e).continuous.measurable.aemeasurable,
    productPotential_unnormalized hH.measurable hG.measurable,productPotential_partition hH.measurable hG.measurable,
    ENNReal.mul_inv (Or.inl (partition_pos_of_continuous hH).ne') (Or.inr (partition_pos_of_continuous hG).ne'),Measure.prod_smul_left,Measure.prod_smul_right,smul_smul]

lemma gaussianPotential_gibbs (d : ℕ) :
    gibbs (fun y : Point d => ‖y‖^2/2)=stdGaussian (Point d) := by
  have he := congrArg (fun μ : Measure (Point d) => μ univ) (EulerDensity.standard_density d)
  have hmul : EulerDensity.gaussianCoefficient d*partition (fun y : Point d => ‖y‖^2/2)=1 := by
    simpa only [measure_univ,Measure.smul_apply,smul_eq_mul,
      withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,partition,gibbsDensity,neg_div] using he.symm
  have hi : (partition (fun y : Point d => ‖y‖^2/2))⁻¹=EulerDensity.gaussianCoefficient d := by
    exact (ENNReal.eq_inv_of_mul_eq_one_left hmul).symm
  rw [EulerDensity.standard_density,gibbs,hi]
  congr 2
  funext y
  simp only [gibbsDensity,neg_div]
end LogConcaveSampling

end

end

end

end OAI