File size: 2,607 Bytes
ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | import OAI.Analysis.FiniteFactor.CrossedProduct.AngleLaw
import Mathlib.Data.Finset.Powerset
namespace OAI
namespace FiniteFactor
open MeasureTheory Finset
open scoped ENNReal Classical
noncomputable section
/-- The finite union event specifying at least q coordinate misses. -/
def coordinateMissEvent (I : Finset SourceCoordinate) (q : ℕ)
(S : SourceCoordinate → Set ℝ) : Set (SourceCoordinate → ℝ) :=
⋃ J ∈ I.powersetCard q, Set.pi (J : Set SourceCoordinate) S
/-- Independence and the union bound, retaining the exact natural exponent.
This lemma uses the actual source angle law. -/
theorem coordinateMissEvent_bound (I : Finset SourceCoordinate) (q : ℕ)
(S : SourceCoordinate → Set ℝ) (hS : ∀ i, MeasurableSet (S i))
(b : ℝ≥0∞) (hb : ∀ i ∈ I, sourceAngleMarginal i (S i) ≤ b) :
sourceAngleLaw (coordinateMissEvent I q S) ≤ (2 : ℝ≥0∞)^I.card * b^q := by
classical
have hJ (J : Finset SourceCoordinate) (hJ : J ∈ I.powersetCard q) :
sourceAngleLaw (Set.pi (J : Set SourceCoordinate) S) ≤ b^q := by
rcases Finset.mem_powersetCard.mp hJ with ⟨hsub, hcard⟩
rw [sourceAngleLaw_cylinder J S (fun i _ => hS i)]
calc
_ ≤ ∏ _i ∈ J, b := Finset.prod_le_prod (fun i hi => hb i (hsub hi))
_ = b^q := by simp [hcard]
have hc : (I.powersetCard q).card ≤ 2^I.card := by
rw [← Finset.card_powerset]
exact Finset.card_le_card (fun J hJ => Finset.mem_powerset.mpr (Finset.mem_powersetCard.mp hJ).1)
calc
_ ≤ ∑ J ∈ I.powersetCard q, sourceAngleLaw (Set.pi (J : Set SourceCoordinate) S) :=
measure_biUnion_finset_le _ _
_ ≤ ∑ _J ∈ I.powersetCard q, b^q := Finset.sum_le_sum hJ
_ = ((I.powersetCard q).card : ℝ≥0∞)*b^q := by simp [nsmul_eq_mul]
_ ≤ (2 : ℝ≥0∞)^I.card*b^q := by
have hc' : ((I.powersetCard q).card : ℝ≥0∞) ≤ (2 : ℝ≥0∞)^I.card := by exact_mod_cast hc
gcongr
/-- Any realization with at least q misses lies in the union event. -/
theorem mem_coordinateMissEvent_of_card (I : Finset SourceCoordinate) (q : ℕ)
(S : SourceCoordinate → Set ℝ) (θ : SourceCoordinate → ℝ)
(h : q ≤ (I.filter (fun i => θ i ∈ S i)).card) : θ ∈ coordinateMissEvent I q S := by
classical
obtain ⟨J, hJ, hcard⟩ := Finset.exists_subset_card_eq h
apply Set.mem_iUnion.mpr
refine ⟨J, Set.mem_iUnion.mpr ⟨Finset.mem_powersetCard.mpr ⟨?_, hcard⟩, ?_⟩⟩
· exact hJ.trans (Finset.filter_subset _ _)
· intro i hi
exact (Finset.mem_filter.mp (hJ hi)).2
end
end FiniteFactor
end OAI
|