| import OAI.NumberTheory.TwoPointCorrelations.CRTComparison |
| import Mathlib.Logic.Equiv.Prod |
|
|
| namespace OAI |
|
|
| /-! Elementary finite averaging identities used to assemble encoded marginals. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
|
|
| noncomputable def uniformAverage {α : Type*} [Fintype α] (f : α → ℝ) : ℝ := |
| (∑ x, f x) / Fintype.card α |
|
|
| @[simp] lemma uniformAverage_const {α : Type*} [Fintype α] [Nonempty α] (c : ℝ) : |
| uniformAverage (fun _ : α => c) = c := by simp [uniformAverage] |
|
|
| lemma uniformAverage_equiv {α β : Type*} [Fintype α] [Fintype β] |
| (e : α ≃ β) (f : β → ℝ) : uniformAverage (fun x => f (e x)) = uniformAverage f := by |
| unfold uniformAverage |
| rw [e.sum_comp, Fintype.card_congr e] |
|
|
| lemma uniformAverage_prod {α β : Type*} [Fintype α] [Fintype β] |
| (f : α → β → ℝ) : |
| uniformAverage (fun x : α × β => f x.1 x.2) = |
| uniformAverage (fun x => uniformAverage (f x)) := by |
| unfold uniformAverage |
| rw [Fintype.sum_prod_type', Fintype.card_prod, Nat.cast_mul, ← Finset.sum_div] |
| rw [div_div] |
| congr 1 |
| ring |
|
|
| /-- A uniform independent family restricted to selected coordinates remains |
| uniform; coordinates outside the selection integrate out exactly. -/ |
| lemma uniformAverage_restrict {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (β : ι → Type*) [∀ i, Fintype (β i)] [∀ i, Nonempty (β i)] (S : Finset ι) |
| (f : (∀ i : S, β i) → ℝ) : |
| uniformAverage (fun x : ∀ i, β i => f (fun i : S => x i)) = uniformAverage f := by |
| let e := Equiv.piEquivPiSubtypeProd (fun i => i ∈ S) β |
| calc |
| _ = uniformAverage (fun x : (∀ i : S, β i) × (∀ i : {i // i ∉ S}, β i) => f x.1) := |
| uniformAverage_equiv e (fun x => f x.1) |
| _ = uniformAverage f := by |
| rw [uniformAverage_prod (fun (x : ∀ i : S, β i) |
| (_y : ∀ i : {i // i ∉ S}, β i) => f x)] |
| simp only [uniformAverage_const] |
|
|
| lemma abs_uniformAverage_le_one {α : Type*} [Fintype α] (f : α → ℝ) |
| (hf : ∀ x, |f x| ≤ 1) : |uniformAverage f| ≤ 1 := by |
| unfold uniformAverage |
| rw [abs_div, show |(Fintype.card α : ℝ)| = (Fintype.card α : ℝ) from |
| abs_of_nonneg (Nat.cast_nonneg _)] |
| calc |
| |∑ x, f x| / (Fintype.card α : ℝ) ≤ |
| (∑ x, |f x|) / (Fintype.card α : ℝ) := |
| div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (Nat.cast_nonneg _) |
| _ ≤ (∑ _x : α, (1 : ℝ)) / (Fintype.card α : ℝ) := |
| div_le_div_of_nonneg_right (Finset.sum_le_sum (fun x _ => hf x)) (Nat.cast_nonneg _) |
| _ ≤ 1 := by simp [div_self_le_one] |
|
|
| /-- Grouping an interval average by its actual CRT residue vector. -/ |
| lemma crtJointProbability_expectation {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (s : ι → ℕ) [∀ i, NeZero (s i)] |
| (hcop : Pairwise (fun i j => (s i).Coprime (s j))) |
| (A : ZMod (∏ i, s i)) (N : ℕ) (F : (∀ i, ZMod (s i)) → ℝ) : |
| (∑ r, crtJointProbability s hcop A N r * F r) = |
| uniformAverage (fun j : Fin N => |
| F (ZMod.prodEquivPi s hcop (A + (j.val : ZMod (∏ i, s i))))) := by |
| simp_rw [crtJointProbability_eq_count, div_mul_eq_mul_div, Finset.sum_mul] |
| rw [← Finset.sum_div, Finset.sum_comm] |
| simp only [ite_mul, one_mul, zero_mul, Finset.sum_ite_eq, Finset.mem_univ, ite_true] |
| unfold uniformAverage |
| rw [Fintype.card_fin] |
| congr 1 |
| exact (Fin.sum_univ_eq_sum_range |
| (fun j : ℕ => F (ZMod.prodEquivPi s hcop (A + (j : ZMod (∏ i, s i))))) N).symm |
|
|
| lemma crtUniform_expectation {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (s : ι → ℕ) [∀ i, NeZero (s i)] (F : (∀ i, ZMod (s i)) → ℝ) : |
| (∑ r, crtUniform s r * F r) = uniformAverage F := by |
| unfold crtUniform uniformAverage |
| rw [← Finset.mul_sum] |
| simp only [Fintype.card_pi, ZMod.card, Nat.cast_prod] |
| ring |
|
|
| /-- Uniform translated interval averages of any bounded observable of the |
| selected residue coordinates differ from the product law by at most `D/N`. -/ |
| theorem crt_observable_difference {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (s : ι → ℕ) [∀ i, NeZero (s i)] |
| (hcop : Pairwise (fun i j => (s i).Coprime (s j))) |
| (A : ZMod (∏ i, s i)) (N : ℕ) (hN : 0 < N) |
| (F : (∀ i, ZMod (s i)) → ℝ) (hF : ∀ r, |F r| ≤ 1) : |
| |uniformAverage (fun j : Fin N => |
| F (ZMod.prodEquivPi s hcop (A + (j.val : ZMod (∏ i, s i))))) - |
| uniformAverage F| ≤ (∏ i, s i : ℕ) / (N : ℝ) := by |
| have h := finite_observable_difference (crtJointProbability s hcop A N) (crtUniform s) F hF |
| rw [crtJointProbability_expectation, crtUniform_expectation] at h |
| calc |
| _ ≤ 2 * finiteTotalVariation (crtJointProbability s hcop A N) (crtUniform s) := h |
| _ = 2 * residueTotalVariation A N := by rw [crtJointProbability_totalVariation] |
| _ ≤ 2 * ((∏ i, s i : ℕ) / (2 * (N : ℝ))) := |
| mul_le_mul_of_nonneg_left (residueTotalVariation_le A N hN) (by norm_num) |
| _ = _ := by ring |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
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|
