| import OAI.Analysis.FiniteFactor.CrossedProduct.AngleLaw |
| import Mathlib.Data.Finset.Powerset |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Finset |
| open scoped ENNReal Classical |
| noncomputable section |
|
|
| / |
| def coordinateMissEvent (I : Finset SourceCoordinate) (q : ℕ) |
| (S : SourceCoordinate → Set ℝ) : Set (SourceCoordinate → ℝ) := |
| ⋃ J ∈ I.powersetCard q, Set.pi (J : Set SourceCoordinate) S |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
