| import OAI.NumberTheory.Ostmann.Arithmetic.BulkProductMixtures |
|
|
| namespace OAI |
|
|
| /-! # The harmonic box masses absorb the joint cell sum -/ |
|
|
| namespace Ostmann |
| open MeasureTheory |
| open scoped Classical BigOperators |
|
|
| /-- Reassembling boxes costs the product of their normalized masses. It does |
| not cost the number of boxes in the main term. -/ |
| theorem bulk_box_sum_bound {J C : Type*} [Fintype J] [Fintype C] |
| (Z : J → ℝ) (hZ : ∀ j, 0 ≤ Z j) (H : J → C → ℝ) |
| (hH : ∀ j c, 0 ≤ H j c) (hmass : ∀ j, Z j * ∑ c, H j c ≤ 2) |
| (B : ℝ) (hB : 0 ≤ B) (E : (J → C) → ℂ) |
| (hE : ∀ c, ‖E c‖ ≤ B * ∏ j, H j (c j)) : |
| ‖((∏ j, Z j : ℝ) : ℂ) * ∑ c, E c‖ ≤ B * 2 ^ Fintype.card J := by |
| have hZp : 0 ≤ ∏ j, Z j := Finset.prod_nonneg fun j _ => hZ j |
| rw [norm_mul, Complex.norm_real, Real.norm_of_nonneg hZp] |
| calc |
| _ ≤ (∏ j, Z j) * ∑ c : J → C, B * ∏ j, H j (c j) := by |
| apply mul_le_mul_of_nonneg_left _ hZp |
| exact (norm_sum_le _ _).trans (Finset.sum_le_sum fun c _ => hE c) |
| _ = B * ∏ j, (Z j * ∑ c, H j c) := by |
| rw [← Finset.mul_sum, ← Fintype.prod_sum, Finset.prod_mul_distrib] |
| ring |
| _ ≤ B * 2 ^ Fintype.card J := by |
| apply mul_le_mul_of_nonneg_left _ hB |
| calc |
| _ ≤ ∏ _j : J, (2 : ℝ) := Finset.prod_le_prod₀ |
| (fun j _ => mul_nonneg (hZ j) (Finset.sum_nonneg fun c _ => hH j c)) |
| (fun j _ => hmass j) |
| _ = _ := by simp only [Finset.prod_const, Finset.card_univ] |
|
|
| /-- The same bound for the exact normalized product of cell measures. -/ |
| theorem BulkIntegrand.integral_product_mixtures_bound {J C : Type*} [Fintype J] [Fintype C] |
| (f : BulkIntegrand J) (Z : J → ℝ) (hZ : ∀ j, 0 ≤ Z j) |
| (μ : J → C → Measure ℝ) [∀ j c, IsFiniteMeasure (μ j c)] |
| (H : J → C → ℝ) (hH : ∀ j c, 0 ≤ H j c) |
| (hmass : ∀ j, Z j * ∑ c, H j c ≤ 2) |
| (B : ℝ) (hB : 0 ≤ B) |
| (hbox : ∀ c : J → C, |
| ‖∫ x, f x ∂Measure.pi (fun j => μ j (c j))‖ ≤ B * ∏ j, H j (c j)) : |
| ‖∫ x, f x ∂Measure.pi (fun j => bulkCellMixture (Z j) (μ j))‖ ≤ B * 2 ^ Fintype.card J := by |
| rw [f.integral_product_mixtures Z hZ μ] |
| exact bulk_box_sum_bound Z hZ H hH hmass B hB _ hbox |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
