lean/OAI/Computability/VertexCover/Analysis/OwnPrivateListsDecoding.lean

import Mathlib
import OAI.Computability.VertexCover.Reduction.DecodingUpperThreeHalves
import OAI.Computability.VertexCover.Analysis.OwnListEnergy

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory

noncomputable def ownPrivateLists (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (s : Φ.BatchWeights J) (β : ℝ) : Φ.PrivateLists d := by
  classical
  exact fun i => if hi : i.1 ∈ J then
    Φ.ownList J frozen c0 A hA ⟨i.1, hi⟩ i (s ⟨i.1, hi⟩) β else []

theorem ownPrivateLists_length (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (s : Φ.BatchWeights J) (m : ℕ) (hm : 0 < m) (i : Φ.Query d) :
    (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m) i).length ≤ Parameters.ell m := by
  classical
  unfold ownPrivateLists
  split_ifs with hi
  · exact Φ.ownList_length J frozen c0 A hA ⟨i.1, hi⟩ i _ m hm
  · simp

theorem ownPrivateLists_membership_measurable (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (β : ℝ)
    (i : Φ.Query d) (a : i.LocalLabel) :
    MeasurableSet {s | a ∈ Φ.ownPrivateLists J frozen c0 A hA s β i} := by
  classical
  unfold ownPrivateLists
  by_cases hi : i.1 ∈ J
  · simp only [dite_eq_left hi, mem_ownList]
    exact measurableSet_le measurable_const
      (((Φ.ownGradient_measurable J frozen c0 A hA ⟨i.1, hi⟩ i (i.slot a)).comp
        (measurable_pi_apply (⟨i.1, hi⟩ : J))).abs)
  · simp only [dite_eq_right hi, List.not_mem_nil, Set.ofPred_false]
    exact MeasurableSet.empty

theorem ownPrivateLists_decoding_upper (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
    (hval : Φ.value ≤ Parameters.σ m)
    (J : Finset (Fin (Parameters.d m))) (hJ : J.card = Parameters.h m)
    (frozen : Φ.Seeds (Parameters.d m)) (c0 : Φ.Coordinate (Parameters.d m) → ℝ)
    (A : Finset (Φ.Coordinate (Parameters.d m) → ℝ)) (hA : A.Nonempty) :
    (∫ s, finiteMean (fun hidden : Φ.HiddenSeeds J =>
      (Φ.listHitMaximum (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
        (Φ.spliceSeeds J frozen hidden) J : ℝ)) ∂Φ.batchLaw J) < 3/2 := by
  classical
  let B : ℝ := 1 + ((Parameters.h m).choose 2 : ℝ) *
    (Parameters.ell m : ℝ)^2 * Parameters.σ m
  have hbound : ∀ s : Φ.BatchWeights J, finiteMean (fun hidden : Φ.HiddenSeeds J =>
      (Φ.listHitMaximum (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
        (Φ.spliceSeeds J frozen hidden) J : ℝ)) ≤ B := by
    intro s
    have hh := Φ.decoding_upper_bound hval
      (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
      (Φ.ownPrivateLists_length J frozen c0 A hA s m (by omega)) J frozen
    simpa only [hJ] using hh
  have hint := integral_mono_of_nonneg (μ := Φ.batchLaw J)
    (f := fun s => finiteMean (fun hidden : Φ.HiddenSeeds J =>
      (Φ.listHitMaximum (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
        (Φ.spliceSeeds J frozen hidden) J : ℝ)))
    (g := fun _ => B) (Filter.Eventually.of_forall (fun s => by
      unfold finiteMean
      exact div_nonneg (Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _))
        (Nat.cast_nonneg _))) (integrable_const B) (Filter.Eventually.of_forall hbound)
  simp only [integral_const, probReal_univ, smul_eq_mul, one_mul] at hint
  have hp := Parameters.pair_budget_lt m hm
  dsimp [B] at hint
  linarith

end VertexCover.LabelCover


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end OAI