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9b6e014 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import Mathlib
import OAI.Probability.BinarySweep.FiniteLaws.UniformFiniteProbability
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.Sparse
variable {I Ω L : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
theorem occupancy_probability_le [Nonempty Ω]
(p : Ω → ℝ) (hp : ∀u, 0 ≤ p u) (hp1 : ∑u, p u=1)
(E : Ω → Prop) (δ : ℝ) (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1)
(hE : finiteProbability p E ≤ δ) (w : ℕ) :
productProbability p (fun x : I → Ω => w ≤ (Finset.univ.filter (fun i => E (x i))).card) ≤
(2:ℝ)^Fintype.card I * δ^w := by
let root : Ω := Classical.choice inferInstance
have hT (T : Finset I) :
productProbability p (fun x : I → Ω => w ≤ T.card ∧ ∀i∈T, E (x i)) ≤ δ^w := by
by_cases h : w ≤ T.card
· have hb := depthOne_probability_le p hp hp1 (fun u _ => E u) δ hδ
(fun _ => hE) T (fun _ => Sum.inr ()) (fun _ : Unit => root)
have he : (fun x : I → Ω => w ≤ T.card ∧ ∀i∈T, E (x i)) =
(fun x : I → Ω => ∀i : T, E (x i)) := by
funext x
simp only [h,true_and,Subtype.forall]
rw [he]
exact hb.trans (pow_le_pow_of_le_one hδ hδ1 h)
· simp only [h,false_and,productProbability,ite_false,mul_zero,Finset.sum_const_zero]
exact pow_nonneg hδ w
calc
_ ≤ productProbability p (fun x : I → Ω => ∃T : Finset I, w ≤ T.card ∧ ∀i∈T, E (x i)) :=
productProbability_mono hp (fun x hx => ⟨_,hx,fun i hi => (Finset.mem_filter.mp hi).2⟩)
_ ≤ ∑T : Finset I, productProbability p (fun x : I → Ω => w ≤ T.card ∧ ∀i∈T, E (x i)) :=
productProbability_union_le hp _
_ ≤ _ := by
simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_finset,nsmul_eq_mul,Nat.cast_pow,Nat.cast_ofNat]
using Finset.sum_le_sum (s:=Finset.univ) (fun T _ => hT T)
variable [Fintype L] [DecidableEq L]
def smallSetEncode {r : ℕ} (S : {S : Finset L // S.card ≤ r}) : Fin r → Option L :=
fun i => if h : i.val < S.val.card then
some ((Fintype.equivFin S.val).symm ⟨i.val,by simpa using h⟩).val else none
omit [Fintype L] [DecidableEq L] in
lemma smallSetEncode_mem {r : ℕ} (S : {S : Finset L // S.card ≤ r}) (l : L) :
l∈S.val ↔ ∃i, smallSetEncode S i=some l := by
constructor
· intro hl
let j := Fintype.equivFin S.val ⟨l,hl⟩
have hj : j.val < S.val.card := by simpa using j.isLt
refine ⟨⟨j.val,lt_of_lt_of_le hj S.property⟩,?_⟩
simp only [smallSetEncode,dite_eq_left hj]
have he : (⟨j.val,by simpa using hj⟩ : Fin (Fintype.card S.val))=j := rfl
rw [he,Equiv.symm_apply_apply]
· rintro ⟨i,hi⟩
unfold smallSetEncode at hi
split_ifs at hi with h
· cases Option.some.inj hi
exact ((Fintype.equivFin S.val).symm ⟨i.val,by simpa using h⟩).property
omit [Fintype L] [DecidableEq L] in
lemma smallSetEncode_injective (r : ℕ) : Function.Injective (smallSetEncode (L:=L) (r:=r)) := by
intro S T h
apply Subtype.ext
ext l
rw [smallSetEncode_mem,smallSetEncode_mem,h]
omit [DecidableEq L] in
lemma smallSet_card_bound (r : ℕ) :
Fintype.card {S : Finset L // S.card ≤ r} ≤ (Fintype.card L+1)^r := by
simpa only [Fintype.card_fun,Fintype.card_option,Fintype.card_fin]
using Fintype.card_le_of_injective _ (smallSetEncode_injective (L:=L) r)
theorem line_sets_occupancy_probability_le [Nonempty Ω]
(p : Ω → ℝ) (hp : ∀u, 0 ≤ p u) (hp1 : ∑u, p u=1)
(line : Ω → L) (r w : ℕ) (δ : ℝ) (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1)
(hmass : ∀S : Finset L, S.card ≤ r → finiteProbability p (fun u => line u∈S) ≤ δ) :
productProbability p (fun x : I → Ω => ∃S : Finset L, S.card ≤ r ∧
w ≤ (Finset.univ.filter (fun i => line (x i)∈S)).card) ≤
((Fintype.card L+1:ℕ):ℝ)^r * (2:ℝ)^Fintype.card I * δ^w := by
let E (S : {S : Finset L // S.card ≤ r}) (x : I → Ω) :=
w ≤ (Finset.univ.filter (fun i => line (x i)∈S.val)).card
have h1 := productProbability_union_le hp E
have h2 : (∑S, productProbability p (E S)) ≤
(Fintype.card {S : Finset L // S.card ≤ r}:ℝ)*((2:ℝ)^Fintype.card I*δ^w) := by
calc
_ ≤ ∑ S : {S : Finset L // S.card ≤ r}, ((2:ℝ)^Fintype.card I*δ^w) := by
apply Finset.sum_le_sum
intro S _
convert occupancy_probability_le p hp hp1 (fun u => line u∈S.val) δ hδ hδ1
(hmass S.val S.property) w (I:=I) using 1
unfold productProbability E
apply Finset.sum_congr rfl
intro x _
congr 2
apply congrArg (fun t : Finset I => w ≤ t.card)
ext i
simp only [Finset.mem_filter,Finset.mem_univ,true_and]
_ = _ := by simp
have h3 : (Fintype.card {S : Finset L // S.card ≤ r}:ℝ) ≤ ((Fintype.card L+1:ℕ):ℝ)^r := by
exact_mod_cast smallSet_card_bound (L:=L) r
have he : (fun x : I → Ω => ∃S : Finset L, S.card ≤ r ∧
w ≤ (Finset.univ.filter (fun i => line (x i)∈S)).card) = (fun x => ∃S, E S x) := by
funext x
simp only [E,Subtype.exists,exists_prop]
rw [he]
apply h1.trans (h2.trans _)
rw [mul_assoc]
exact mul_le_mul_of_nonneg_right h3 (mul_nonneg (pow_nonneg (by norm_num) _) (pow_nonneg hδ w))
end BinaryCoordinateSweeps.Sparse
end
end OAI
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