lean/OAI/Probability/Perceptron/Variational/CountableGaussianResampling.lean

import Mathlib
import OAI.Probability.Perceptron.Brownian.CountableGaussianField

namespace OAI

noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators
namespace SphericalPerceptronFreeEnergy

 def resampleGaussianCoordinate (i : ℕ) (p : ℝ×(ℕ → ℝ)) : ℕ → ℝ :=
  Function.update p.2 i p.1

lemma resampleGaussianCoordinate_measurable (i : ℕ) :
    Measurable (resampleGaussianCoordinate i) := by
  unfold resampleGaussianCoordinate
  fun_prop

lemma resampleGaussianCoordinate_measurePreserving (i : ℕ) :
    MeasurePreserving (resampleGaussianCoordinate i)
      ((gaussianReal 0 1).prod countableGaussianLaw) countableGaussianLaw := by
  refine ⟨resampleGaussianCoordinate_measurable i,?_⟩
  apply Measure.eq_infinitePi
  intro s t ht
  rw [Measure.map_apply (resampleGaussianCoordinate_measurable i)
    (MeasurableSet.pi s.countable_toSet (fun j _ => ht j))]
  classical
  by_cases hi : i ∈ s
  · have he : (resampleGaussianCoordinate i) ⁻¹' Set.pi s t =
        t i ×ˢ Set.pi (s.erase i) t := by
      ext p
      simp only [mem_preimage,mem_prod,Set.mem_pi,Finset.mem_coe,resampleGaussianCoordinate]
      constructor
      · intro hp
        refine ⟨?_,fun j hj => ?_⟩
        · simpa only [Function.update_self] using hp i hi
        · simpa only [Function.update_of_ne (Finset.mem_erase.mp hj).1] using
            hp j (Finset.mem_of_mem_erase hj)
      · rintro ⟨hp,hq⟩ j hj
        by_cases he : j=i
        · subst j
          simpa only [Function.update_self] using hp
        · simpa only [Function.update_of_ne he] using hq j (Finset.mem_erase.mpr ⟨he,hj⟩)
    rw [he,Measure.prod_prod,countableGaussianLaw,Measure.infinitePi_pi _ (fun j _ => ht j)]
    exact Finset.mul_prod_erase s (fun j => gaussianReal 0 1 (t j)) hi
  · have he : (resampleGaussianCoordinate i) ⁻¹' Set.pi s t =
        Set.univ ×ˢ Set.pi s t := by
      ext p
      simp only [mem_preimage,mem_prod,true_and,mem_univ,Set.mem_pi,Finset.mem_coe,resampleGaussianCoordinate]
      constructor <;> intro hp j hj
      · simpa only [Function.update_of_ne (ne_of_mem_of_not_mem hj hi)] using hp j hj
      · simpa only [Function.update_of_ne (ne_of_mem_of_not_mem hj hi)] using hp j hj
    rw [he,Measure.prod_prod,measure_univ,one_mul,countableGaussianLaw,
      Measure.infinitePi_pi _ (fun j _ => ht j)]

end SphericalPerceptronFreeEnergy
end

end OAI