| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped BigOperators | |
| namespace SparseWitness | |
| attribute [local instance] Classical.propDecidable | |
| lemma joint_witness_bound {Ω E I : Type*} [MeasurableSpace Ω] | |
| [Fintype I] [DecidableEq E] {T : I → Type*} [∀ i, Fintype (T i)] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (test : E → Set Ω) | |
| (edges : ∀ i, T i → Finset E) (q : ℝ) (_hq : 0 ≤ q) | |
| (joint : ∀ F : Finset E, μ.real {ω | ∀ e ∈ F, ω ∈ test e} ≤ q^F.card) | |
| (disj : ∀ i j, i ≠ j → ∀ s t, Disjoint (edges i s) (edges j t)) : | |
| μ.real {ω | ∀ i, ∃ t, ∀ e ∈ edges i t, ω ∈ test e} ≤ | |
| ∏ i, ∑ t, q^(edges i t).card := by | |
| classical | |
| have hsub : {ω | ∀ i, ∃ t, ∀ e ∈ edges i t, ω ∈ test e} ⊆ | |
| ⋃ t : ∀ i, T i, {ω | ∀ e ∈ Finset.univ.biUnion (fun i => edges i (t i)), ω ∈ test e} := by | |
| intro ω hω | |
| choose t ht using hω | |
| refine Set.mem_iUnion.mpr ⟨t,?_⟩ | |
| intro e he | |
| obtain ⟨i,hi,he⟩ := Finset.mem_biUnion.mp he | |
| exact ht i e he | |
| calc | |
| _ ≤ μ.real (⋃ t : ∀ i, T i, {ω | ∀ e ∈ Finset.univ.biUnion (fun i => edges i (t i)), ω ∈ test e}) := | |
| measureReal_mono hsub | |
| _ ≤ ∑ t : ∀ i, T i, μ.real {ω | ∀ e ∈ Finset.univ.biUnion (fun i => edges i (t i)), ω ∈ test e} := | |
| measureReal_iUnion_fintype_le _ | |
| _ ≤ ∑ t : ∀ i, T i, q^(Finset.univ.biUnion (fun i => edges i (t i))).card := | |
| Finset.sum_le_sum (fun t _ => joint _) | |
| _ = ∑ t : ∀ i, T i, ∏ i, q^(edges i (t i)).card := by | |
| apply Finset.sum_congr rfl | |
| intro t _ | |
| rw [Finset.card_biUnion (fun i _ j _ hij => disj i j hij (t i) (t j))] | |
| exact (Finset.prod_pow_eq_pow_sum _ _ _).symm | |
| _ = _ := by simpa using (Finset.prod_univ_sum (fun i => (Finset.univ : Finset (T i))) | |
| (fun i t => q^(edges i t).card)).symm | |
| lemma large_fiber {I J : Type*} [Fintype I] [Fintype J] [Nonempty J] | |
| (f : I → J) : ∃ j, Fintype.card I ≤ Fintype.card J * | |
| (Finset.univ.filter (fun i => f i = j)).card := by | |
| classical | |
| obtain ⟨j,hj,hmax⟩ := Finset.exists_max_image Finset.univ | |
| (fun j => (Finset.univ.filter (fun i => f i = j)).card) Finset.univ_nonempty | |
| refine ⟨j,?_⟩ | |
| calc | |
| Fintype.card I = ∑ c : J, (Finset.univ.filter (fun i => f i = c)).card := by | |
| simpa using (Finset.card_eq_sum_card_fiberwise (s := (Finset.univ : Finset I)) | |
| (t := Finset.univ) (f := f) (fun _ _ => Finset.mem_univ _)) | |
| _ ≤ ∑ _c : J, (Finset.univ.filter (fun i => f i = j)).card := | |
| Finset.sum_le_sum (fun c hc => hmax c hc) | |
| _ = _ := by simp | |
| lemma alternative_sparse_bound {Ω I J : Type*} [MeasurableSpace Ω] | |
| [Fintype I] [Fintype J] [Nonempty J] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (test : J → I → Set Ω) | |
| (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) | |
| (joint : ∀ j (A : Finset I), μ.real {ω | ∀ i ∈ A, ω ∈ test j i} ≤ q^A.card) : | |
| μ.real {ω | ∀ i, ∃ j, ω ∈ test j i} ≤ | |
| (Fintype.card J : ℝ)^(Fintype.card I) * | |
| q^((Fintype.card I : ℝ)/(Fintype.card J : ℝ)) := by | |
| classical | |
| have hsub : {ω | ∀ i, ∃ j, ω ∈ test j i} ⊆ | |
| ⋃ f : I → J, {ω | ∀ i, ω ∈ test (f i) i} := by | |
| intro ω hω | |
| choose f hf using hω | |
| exact Set.mem_iUnion.mpr ⟨f,hf⟩ | |
| have each (f : I → J) : μ.real {ω | ∀ i, ω ∈ test (f i) i} ≤ | |
| q^((Fintype.card I : ℝ)/(Fintype.card J : ℝ)) := by | |
| obtain ⟨j,hj⟩ := large_fiber f | |
| let A := Finset.univ.filter (fun i => f i = j) | |
| have hb : {ω | ∀ i, ω ∈ test (f i) i} ⊆ {ω | ∀ i ∈ A, ω ∈ test j i} := by | |
| intro ω hω i hi | |
| simpa only [(Finset.mem_filter.mp hi).2] using hω i | |
| have hJ : (0 : ℝ) < Fintype.card J := Nat.cast_pos.mpr Fintype.card_pos | |
| have hsize : (Fintype.card I : ℝ)/(Fintype.card J : ℝ) ≤ A.card := by | |
| apply (div_le_iff₀ hJ).mpr | |
| have hh : (Fintype.card I : ℝ) ≤ (Fintype.card J : ℝ)*A.card := by exact_mod_cast hj | |
| nlinarith | |
| calc | |
| _ ≤ μ.real {ω | ∀ i ∈ A, ω ∈ test j i} := measureReal_mono hb | |
| _ ≤ q^A.card := joint j A | |
| _ = q^(A.card : ℝ) := (Real.rpow_natCast _ _).symm | |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_ge hq hq1 hsize | |
| calc | |
| _ ≤ μ.real (⋃ f : I → J, {ω | ∀ i, ω ∈ test (f i) i}) := measureReal_mono hsub | |
| _ ≤ ∑ f : I → J, μ.real {ω | ∀ i, ω ∈ test (f i) i} := measureReal_iUnion_fintype_le _ | |
| _ ≤ ∑ _ : I → J, q^((Fintype.card I : ℝ)/(Fintype.card J : ℝ)) := | |
| Finset.sum_le_sum (fun f _ => each f) | |
| _ = _ := by simp | |
| end SparseWitness | |
| end | |
| end OAI | |