import Mathlib
import OAI.Computability.VertexCover.Analysis.OwnTotalEnergyNo
import OAI.Computability.VertexCover.Analysis.Poincare

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory

theorem ownMean_poincare (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)
    (j : J) (i : Φ.Query d) :
    variance (Φ.ownMean J frozen c0 A hA j i)
      (VertexCover.Cube.law (Fin (Φ.WeightDimension d))) ≤
      2 * ∫ s, ∑ k, (Φ.ownGradient J frozen c0 A hA j i s k)^2
        ∂VertexCover.Cube.law (Fin (Φ.WeightDimension d)) := by
  apply VertexCover.Cube.poincare (Φ.ownMean_lipschitz J frozen c0 A hA j i)
    (Φ.ownGradient J frozen c0 A hA j i)
    (Φ.ownGradient_measurable J frozen c0 A hA j i)
  · intro s k
    exact VertexCover.Threshold.coordinate_bound (Φ.ownGradient_block_bound J frozen c0 A hA j i s) k
  · exact Φ.ownMean_partial_ae J frozen c0 A hA j i

noncomputable def ownTotalVariance (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) : ℝ :=
  ∑ j : J, VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J =>
    variance (Φ.ownMean J frozen c0 A hA j (Φ.query (Φ.spliceSeeds J frozen hidden) j))
      (VertexCover.Cube.law (Fin (Φ.WeightDimension d))))

theorem ownGradient_sq_integrable (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)
    (j : J) (i : Φ.Query d) :
    Integrable (fun s => ∑ k, (Φ.ownGradient J frozen c0 A hA j i s k)^2)
      (VertexCover.Cube.law (Fin (Φ.WeightDimension d))) := by
  apply integrable_finsetSum
  intro k hk
  apply Integrable.of_bound ((Φ.ownGradient_measurable J frozen c0 A hA j i k).pow_const 2).aestronglyMeasurable 1
  filter_upwards [] with s
  have hb := VertexCover.Threshold.coordinate_bound (Φ.ownGradient_block_bound J frozen c0 A hA j i s) k
  rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
  nlinarith [abs_nonneg (Φ.ownGradient J frozen c0 A hA j i s k),
    sq_abs (Φ.ownGradient J frozen c0 A hA j i s k)]

theorem ownTotalVariance_le_energy (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) :
    Φ.ownTotalVariance J frozen c0 A hA ≤ 2 * Φ.ownTotalEnergy J frozen c0 A hA := by
  classical
  unfold ownTotalVariance ownTotalEnergy
  rw [Finset.mul_sum]
  apply Finset.sum_le_sum
  intro j hj
  rw [VertexCover.integral_finiteMean _ _ (fun hidden =>
    Φ.ownGradient_sq_integrable J frozen c0 A hA j (Φ.query (Φ.spliceSeeds J frozen hidden) j))]
  calc
    _ ≤ VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J =>
      2 * ∫ s, ∑ k, (Φ.ownGradient J frozen c0 A hA j
        (Φ.query (Φ.spliceSeeds J frozen hidden) j) s k)^2
          ∂VertexCover.Cube.law (Fin (Φ.WeightDimension d))) :=
      VertexCover.finiteMean_mono (fun hidden => Φ.ownMean_poincare J frozen c0 A hA j _)
    _ = _ := by simp only [VertexCover.finiteMean, ← Finset.mul_sum]; ring

theorem ownTotalVariance_no_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) :
    Φ.ownTotalVariance J frozen c0 A hA < 7 := by
  have he := Φ.ownTotalVariance_le_energy J frozen c0 A hA
  have hu := Φ.ownTotalEnergy_no_upper m hm hval J hJ frozen c0 A hA
  linarith

end VertexCover.LabelCover


end
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end

end OAI