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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
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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