lean/OAI/Computability/VertexCover/Analysis/IntegralUpdate.lean

import Mathlib
import OAI.Computability.VertexCover.Fourier.SourceOccurrences
import OAI.Computability.VertexCover.Analysis.OwnPrivateListsDecoding

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory

theorem listHitMaximum_measurable {E : Type*} [MeasurableSpace E]
    (Φ : LabelCover) {d : ℕ} (L : E → Φ.PrivateLists d)
    (hL : ∀ i (a : i.LocalLabel), MeasurableSet {x | a ∈ L x i})
    (seed : Φ.Seeds d) (J : Finset (Fin d)) :
    Measurable (fun x => Φ.listHitMaximum (L x) seed J) := by
  classical
  have hhit : ∀ I : Finset (Φ.Coordinate d), ∀ j : Fin d,
      MeasurableSet {x | Φ.ListHit (L x) seed I j} := by
    intro I j
    change MeasurableSet {x | ∃ a : (Φ.query seed j).LocalLabel,
      a ∈ L x (Φ.query seed j) ∧ (⟨Φ.query seed j, a⟩ : Φ.Coordinate d) ∈ I}
    simp only [Set.ofPred_exists]
    apply MeasurableSet.iUnion
    intro a
    exact (hL _ a).inter (MeasurableSet.const _)
  have hcard : ∀ I : Finset (Φ.Coordinate d),
      Measurable (fun x => (Φ.hitPositions (L x) seed I J).card) := by
    intro I
    simp only [hitPositions, Finset.card_filter]
    apply Finset.measurable_sum
    intro j _
    exact Measurable.ite (hhit I j) measurable_const measurable_const
  have hsup : ∀ S : Finset (Finset (Φ.Coordinate d)),
      Measurable (fun x => S.sup (fun I => (Φ.hitPositions (L x) seed I J).card)) := by
    intro S
    induction S using Finset.induction_on with
    | empty => simp
    | @insert I S hi ih =>
      simp only [Finset.sup_insert]
      exact (hcard I).sup ih
  exact hsup _

theorem ownPrivateLists_hits_integrable (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (frozen seed : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (β : ℝ) :
    Integrable (fun s => (Φ.listHitMaximum
      (Φ.ownPrivateLists J frozen c0 A hA s β) seed J : ℝ)) (Φ.batchLaw J) := by
  classical
  have hmeas := Φ.listHitMaximum_measurable
    (fun s => Φ.ownPrivateLists J frozen c0 A hA s β)
    (Φ.ownPrivateLists_membership_measurable J frozen c0 A hA β) seed J
  have hcast : Measurable (fun s => (Φ.listHitMaximum
      (Φ.ownPrivateLists J frozen c0 A hA s β) seed J : ℝ)) :=
    (measurable_of_countable (fun n : ℕ => (n : ℝ))).comp hmeas
  apply Integrable.of_bound hcast.aestronglyMeasurable (J.card : ℝ)
  filter_upwards [] with s
  rw [Real.norm_eq_abs, abs_of_nonneg (Nat.cast_nonneg _)]
  apply Nat.cast_le.mpr
  apply Finset.sup_le
  intro I _
  exact Finset.card_le_card (Finset.filter_subset _ _)

end VertexCover.LabelCover

namespace VertexCover.Product
open MeasureTheory

theorem update_measurePreserving {ι E : Type*} [Fintype ι] [DecidableEq ι]
    [MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι) :
    MeasurePreserving (fun p : E × (ι → E) => Function.update p.2 j p.1)
      (μ.prod (Measure.pi (fun _ : ι => μ))) (Measure.pi (fun _ : ι => μ)) := by
  have hm : Measurable (fun p : E × (ι → E) => Function.update p.2 j p.1) :=
    Measurable.of_eval (fun k => by
      by_cases hk : k = j
      · subst k; simpa using measurable_fst
      · simpa only [Function.update_of_ne hk, Function.comp_def] using
          (measurable_pi_apply k).comp measurable_snd)
  refine ⟨hm, (Measure.pi_eq (fun S hS => ?_)).symm⟩
  rw [Measure.map_apply hm (MeasurableSet.univ_pi hS)]
  have hpre : (fun p : E × (ι → E) => Function.update p.2 j p.1) ⁻¹' Set.univ.pi S =
      S j ×ˢ Set.univ.pi (Function.update S j Set.univ) := by
    ext p
    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_prod]
    constructor
    · intro hp
      refine ⟨?_, ?_⟩
      · simpa using hp j
      · intro k
        by_cases hk : k = j
        · subst k; simp
        · simpa [Function.update_of_ne hk] using hp k
    · rintro ⟨hp, hq⟩ k
      by_cases hk : k = j
      · subst k; simpa using hp
      · simpa [Function.update_of_ne hk] using hq k
  rw [hpre, Measure.prod_prod, Measure.pi_pi]
  have hfun : (fun k => μ (Function.update S j Set.univ k)) =
      Function.update (fun k => μ (S k)) j 1 := by
    funext k
    by_cases hk : k = j <;> simp [hk]
  rw [hfun, Finset.prod_update_of_mem (Finset.mem_univ j), one_mul]
  simpa only [Finset.sdiff_singleton_eq_erase] using
    Finset.mul_prod_erase Finset.univ (fun k => μ (S k)) (Finset.mem_univ j)

theorem update_ae {ι E : Type*} [Fintype ι] [DecidableEq ι]
    [MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι)
    {P : (ι → E) → Prop} (hP : ∀ᵐ s ∂Measure.pi (fun _ : ι => μ), P s) :
    ∀ᵐ x ∂μ, ∀ᵐ s ∂Measure.pi (fun _ : ι => μ), P (Function.update s j x) := by
  exact Measure.ae_ae_of_ae_prod ((update_measurePreserving μ j).quasiMeasurePreserving.ae hP)

theorem integral_update {ι E : Type*} [Fintype ι] [DecidableEq ι]
    [MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι)
    {f : (ι → E) → ℝ} (hf : Integrable f (Measure.pi (fun _ : ι => μ))) :
    (∫ x, ∫ s, f (Function.update s j x) ∂Measure.pi (fun _ : ι => μ) ∂μ) =
      ∫ s, f s ∂Measure.pi (fun _ : ι => μ) := by
  have hp := update_measurePreserving μ j
  calc
    _ = ∫ p, f (Function.update p.2 j p.1)
        ∂(μ.prod (Measure.pi (fun _ : ι => μ))) := by
      simpa only [Function.comp_apply] using
        (integral_prod _ (hp.integrable_comp_of_integrable hf)).symm
    _ = ∫ s, f s ∂Measure.map (fun p : E × (ι → E) => Function.update p.2 j p.1)
        (μ.prod (Measure.pi (fun _ : ι => μ))) := by
      exact (integral_map hp.measurable.aemeasurable
        (by simpa only [hp.map_eq] using hf.aestronglyMeasurable)).symm
    _ = _ := by rw [hp.map_eq]

end VertexCover.Product


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end OAI