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dd14c26 | 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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
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