lean/OAI/Probability/CriticalPercolation/ProductMeasure/BitMeasure.lean

import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Density

namespace OAI

section
section
open scoped BigOperators
open Set MeasureTheory ProbabilityTheory
open scoped Classical ENNReal

namespace CriticalPercolation
namespace BernoulliProduct

noncomputable def bitMeasure (p : unitInterval) : Measure Prop :=
  unitInterval.toNNReal p • Measure.dirac True +
    unitInterval.toNNReal (unitInterval.symm p) • Measure.dirac False

instance bitMeasure_probability (p : unitInterval) : IsProbabilityMeasure (bitMeasure p) := by
  constructor
  simp [bitMeasure]

noncomputable def bitWeight (p : unitInterval) (b : Prop) : ℝ :=
  if b then p else 1-p

lemma bitWeight_sum (p : unitInterval) : ∑ b, bitWeight p b = 1 := by
  simp [bitWeight]

lemma bitWeight_nonneg (p : unitInterval) (b : Prop) : 0 ≤ bitWeight p b := by
  unfold bitWeight
  split_ifs
  · exact p.property.1
  · exact sub_nonneg.mpr p.property.2

lemma bitMeasure_real_singleton (p : unitInterval) (b : Prop) :
    (bitMeasure p).real {b} = bitWeight p b := by
  by_cases hb : b
  · have he : b = True := propext (iff_true_intro hb)
    subst b
    simp [bitMeasure, bitWeight, measureReal_def]
  · have he : b = False := propext (iff_false_intro hb)
    subst b
    simp [bitMeasure, bitWeight, measureReal_def]

variable {E : Type*}

noncomputable def law (q : E → unitInterval) : Measure (Set E) :=
  Measure.map MeasurableEquiv.setOfPred (Measure.infinitePi fun e => bitMeasure (q e))

instance law_probability (q : E → unitInterval) : IsProbabilityMeasure (law q) := by
  unfold law
  infer_instance

lemma law_const (p : unitInterval) :
    law (fun _ : E => p) = setBernoulli Set.univ p := by
  rw [law, setBernoulli_eq_map]
  rfl

lemma integral_bits_eq_expect [Fintype E] (q : E → unitInterval) (f : (E → Prop) → ℝ) :
    (∫ ω, f ω ∂(Measure.infinitePi fun e => bitMeasure (q e))) =
      FiniteProduct.expect (fun e => bitWeight (q e)) f := by
  rw [Measure.infinitePi_eq_pi, integral_fintype Integrable.of_finite]
  unfold FiniteProduct.expect FiniteProduct.density
  apply Finset.sum_congr rfl
  intro ω _
  rw [smul_eq_mul]
  congr 1
  simp only [measureReal_def, Measure.pi_singleton, ENNReal.toReal_prod]
  exact Finset.prod_congr rfl (fun e _ => bitMeasure_real_singleton (q e) (ω e))

lemma integral_law_eq_expect [Fintype E] (q : E → unitInterval) (f : Set E → ℝ) :
    (∫ ω, f ω ∂law q) =
      FiniteProduct.expect (fun e => bitWeight (q e)) (fun ω => f {e | ω e}) := by
  rw [law, integral_map MeasurableEquiv.setOfPred.measurable.aemeasurable
    Integrable.of_finite.aestronglyMeasurable, integral_bits_eq_expect]
  rfl

lemma law_map_restrict (q : E → unitInterval) (s : Set E) :
    (law q).map (fun ω : Set E => Subtype.val ⁻¹' ω : Set E → Set s) =
      law (fun e : s => q e) := by
  have hmeas : Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) :=
    measurable_set_iff.mpr (fun bond => measurable_set_mem bond.val)
  unfold law
  rw [Measure.map_map hmeas MeasurableEquiv.setOfPred.measurable]
  have hcomp : (fun ω : Set E => Subtype.val ⁻¹' ω : Set E → Set s) ∘
      MeasurableEquiv.setOfPred =
      MeasurableEquiv.setOfPred ∘ s.domRestrict := rfl
  rw [hcomp, ← Measure.map_map MeasurableEquiv.setOfPred.measurable (by fun_prop),
    Measure.infinitePi_map_restrict']

end BernoulliProduct
end CriticalPercolation
open Set
open scoped Classical

end
end

end OAI