import Mathlib
import OAI.Computability.VertexCover.Reduction.PairEventMean

namespace OAI

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namespace VertexCover.Average
open MeasureTheory

noncomputable def fiberAverage {α β : Type*} [Fintype α]
    (q : α → β) (f : β → α → ℝ) (i : β) : ℝ := by
  classical
  exact (∑ a ∈ Finset.univ.filter (fun a => q a = i), f i a) /
    (Finset.univ.filter (fun a => q a = i)).card

theorem sum_fiberAverage {α β : Type*} [Fintype α] [Fintype β]
    (q : α → β) (f : β → α → ℝ) :
    (∑ a, fiberAverage q f (q a)) = ∑ a, f (q a) a := by
  classical
  rw [← Finset.sum_fiberwise Finset.univ q (fun a => fiberAverage q f (q a)),
    ← Finset.sum_fiberwise Finset.univ q (fun a => f (q a) a)]
  apply Finset.sum_congr rfl
  intro i hi
  have hq : ∀ a ∈ Finset.univ.filter (fun a => q a = i), q a = i := by
    intro a ha; exact (Finset.mem_filter.mp ha).2
  calc
    _ = ∑ _a ∈ Finset.univ.filter (fun a => q a = i), fiberAverage q f i := by
      apply Finset.sum_congr rfl
      intro a ha
      rw [hq a ha]
    _ = (Finset.univ.filter (fun a => q a = i)).card * fiberAverage q f i := by simp
    _ = ∑ a ∈ Finset.univ.filter (fun a => q a = i), f i a := by
      unfold fiberAverage
      by_cases hz : (Finset.univ.filter (fun a => q a = i)).card = 0
      · have he := Finset.card_eq_zero.mp hz
        simp [he]
      · exact mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr hz)
    _ = _ := by
      apply Finset.sum_congr rfl
      intro a ha
      rw [hq a ha]

theorem finiteMean_fiber_tower {α β : Type*} [Fintype α] [Fintype β]
    (q : α → β) (f : β → α → ℝ) :
    (by classical exact VertexCover.finiteMean (fun a : α =>
      VertexCover.finiteMean (fun a' : Finset.univ.filter (fun a' => q a' = q a) =>
        f (q a) a'))) = VertexCover.finiteMean (fun a : α => f (q a) a) := by
  classical
  have he : ∀ i, VertexCover.finiteMean (fun a : Finset.univ.filter (fun a => q a = i) => f i a) =
      fiberAverage q f i := by
    intro i
    unfold VertexCover.finiteMean fiberAverage
    rw [Finset.sum_coe_sort, Fintype.card_coe]
  simp_rw [he]
  unfold VertexCover.finiteMean
  rw [sum_fiberAverage]

theorem finiteMean_fiber_tower_of_mem {α β : Type*} [Fintype α] [Fintype β]
    (q : α → β) (f : β → α → ℝ) (T : β → Finset α)
    (hT : ∀ i a, a ∈ T i ↔ q a = i) :
    VertexCover.finiteMean (fun a : α =>
      VertexCover.finiteMean (fun a' : T (q a) => f (q a) a')) =
      VertexCover.finiteMean (fun a : α => f (q a) a) := by
  classical
  have he : T = fun i => Finset.univ.filter (fun a => q a = i) := by
    funext i
    ext a
    simp only [Finset.mem_filter, Finset.mem_univ, true_and, hT]
  subst T
  exact finiteMean_fiber_tower q f

theorem selected_integral_abs_le {α Ω K : Type*} [Fintype α] [MeasurableSpace Ω]
    (μ : Measure Ω) (T : Finset K) (F : α → Ω → K → ℝ)
    (hF : ∀ a k, Integrable (fun x => F a x k) μ) :
    (∑ k ∈ T, |(∑ a, ∫ x, F a x k ∂μ) / Fintype.card α|) ≤
      (∑ a, ∫ x, ∑ k ∈ T, |F a x k| ∂μ) / Fintype.card α := by
  classical
  have hc : (0 : ℝ) ≤ Fintype.card α := Nat.cast_nonneg _
  simp_rw [abs_div, abs_of_nonneg hc]
  rw [← Finset.sum_div]
  apply div_le_div_of_nonneg_right _ hc
  calc
    _ ≤ ∑ k ∈ T, ∑ a, ∫ x, |F a x k| ∂μ := by
      apply Finset.sum_le_sum
      intro k hk
      calc
        _ ≤ ∑ a, |∫ x, F a x k ∂μ| := Finset.abs_sum_le_sum_abs _ _
        _ ≤ _ := Finset.sum_le_sum (fun a _ => by
          simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (F a · k))
    _ = ∑ a, ∑ k ∈ T, ∫ x, |F a x k| ∂μ := Finset.sum_comm
    _ = _ := by
      apply Finset.sum_congr rfl
      intro a ha
      exact (integral_finsetSum T (fun k _ => (hF a k).abs)).symm

end VertexCover.Average


end
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end OAI