| 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 |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
| end |
| end |
| |
| end OAI |
| |
