import Mathlib
import OAI.Computability.VertexCover.Reduction.PrivateTotalEnergyLaw
import OAI.Computability.VertexCover.Reduction.PrivateProjectionBatchLower

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators

theorem restrictionFunction_energy_variance (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    (Φ.restrictionForm d).energy (Φ.restrictionFunction A hA) =
      VertexCover.finiteMean (fun seed : Φ.Seeds d =>
        variance (fun s => Φ.separator A hA (Φ.continuousSum seed s))
          (VertexCover.Cube.blockLaw (Fin d) (Fin (Φ.WeightDimension d)))) := by
  rw [restrictionForm_energy]
  apply congrArg VertexCover.finiteMean
  funext seed
  rw [variance_eq_integral
    (Φ.separator_continuousSum_continuous A hA seed).measurable.aemeasurable,
    Φ.separator_continuousSum_mean_zero]
  simp only [sub_zero, restrictionFunction_apply]
  exact CompactCube.integral_realWeights
    ((Φ.separator_continuousSum_continuous A hA seed).pow 2).measurable.aestronglyMeasurable

theorem restriction_frozen_exists (Φ : LabelCover) {d h : ℕ}
    (hd : 4 ≤ d) (hh : 3 ≤ h) (hhd : h ≤ d) (ν : ℝ) (hν : 0 < ν)
    (hscale : 16*(h:ℝ)+32*(h:ℝ)^2 ≤ ν*d/2)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (henergy : ν*d ≤ (Φ.restrictionForm d).energy (Φ.restrictionFunction A hA)) :
    ∃ J : Finset (Fin d), J.card = h ∧ ∃ frozen : Φ.Seeds d,
      ∃ s : Φ.RestrictionWeights d,
        ν*h/4 ≤ Φ.ownTotalVariance J frozen
          (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA := by
  classical
  have : Nonempty (Fin Φ.M) := ⟨⟨0, Φ.M_pos⟩⟩
  by_contra hno
  push Not at hno
  have hbound (J : Finset (Fin d)) (hJ : J ∈ batches d h) :
      (∑ j ∈ J, (Φ.restrictionForm d).energy
        (Φ.privateProjection J j (Φ.restrictionFunction A hA))) ≤ ν*h/4 := by
    rw [← Finset.sum_coe_sort J, Φ.privateTotal_energy_law J A hA]
    calc
      _ ≤ VertexCover.finiteMean (fun _ : Φ.Seeds d => ν*h/4) := by
        apply VertexCover.finiteMean_mono
        intro frozen
        have hb := integral_mono (Φ.ownTotalVariance_frozen_integrable J frozen A hA)
          (integrable_const (ν*h/4))
          (fun s => (hno J (mem_batches.mp hJ) frozen s).le)
        simpa using hb
      _ = _ := VertexCover.finiteMean_const _
  have hsum := Finset.sum_le_sum hbound
  have hl := Φ.privateProjection_batch_lower hd hh ν hscale A hA henergy
  simp only [Finset.sum_const, nsmul_eq_mul, card_batches] at hsum
  have hpos : 0 < (d.choose h:ℝ)*ν*h :=
    mul_pos (mul_pos (Nat.cast_pos.mpr (Nat.choose_pos hhd)) hν)
      (Nat.cast_pos.mpr (by omega))
  linarith

end VertexCover.LabelCover


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end OAI