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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Probability.FirstPassage.BKFamilies
import Mathlib.Basic.Real.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Positivity
namespace OAI
namespace PlanarFPP
open Finset
variable {α : Type*} [DecidableEq α]
lemma upward_disjointOccurrence (s : Finset α) (A B : Finset (Finset α)) :
UpwardFamily s (disjointOccurrenceFamily s A B) := by
constructor
· intro x hx
exact ((mem_disjointOccurrenceFamily ..).mp hx).1
· intro x hx y hxy hys
obtain ⟨_,a,ha,b,hb,hax,hbx,hab⟩ := (mem_disjointOccurrenceFamily ..).mp hx
exact (mem_disjointOccurrenceFamily ..).mpr ⟨hys,a,ha,b,hb,hax.trans hxy,hbx.trans hxy,hab⟩
noncomputable def fairFamilyMass (s : Finset α) (A : Finset (Finset α)) : ℝ :=
A.card / (2:ℝ)^s.card
omit [DecidableEq α] in
lemma fairFamilyMass_nonneg (s : Finset α) (A : Finset (Finset α)) :
0 ≤ fairFamilyMass s A := by unfold fairFamilyMass; positivity
omit [DecidableEq α] in
lemma fairFamilyMass_mono (s : Finset α) {A B : Finset (Finset α)} (h : A ⊆ B) :
fairFamilyMass s A ≤ fairFamilyMass s B := by
exact div_le_div_of_nonneg_right (by exact_mod_cast card_le_card h) (by positivity)
omit [DecidableEq α] in
lemma fairFamilyMass_powerset (s : Finset α) : fairFamilyMass s s.powerset = 1 := by
simp [fairFamilyMass,Finset.card_powerset]
lemma finite_fair_BK_mass (s : Finset α) (A B : Finset (Finset α))
(hA : UpwardFamily s A) (hB : UpwardFamily s B) :
fairFamilyMass s (disjointOccurrenceFamily s A B) ≤ fairFamilyMass s A * fairFamilyMass s B := by
have h := finite_fair_BK s A B hA hB
have hr : (2:ℝ)^s.card * (disjointOccurrenceFamily s A B).card ≤ (A.card:ℝ)*B.card := by
exact_mod_cast h
unfold fairFamilyMass
rw [div_mul_div_comm,div_le_div_iff₀ (by positivity) (by positivity)]
nlinarith [pow_pos (by norm_num : (0:ℝ)<2) s.card]
noncomputable def iteratedDisjointOccurrence (s : Finset α) : List (Finset (Finset α)) → Finset (Finset α)
| [] => s.powerset
| A :: As => disjointOccurrenceFamily s A (iteratedDisjointOccurrence s As)
lemma upward_iteratedDisjointOccurrence (s : Finset α) (As : List (Finset (Finset α))) :
UpwardFamily s (iteratedDisjointOccurrence s As) := by
cases As with
| nil =>
constructor
· intro x hx; exact mem_powerset.mp hx
· intro x hx y hxy hys; exact mem_powerset.mpr hys
| cons A As => exact upward_disjointOccurrence s A _
theorem finite_fair_BK_repeated (s : Finset α) (As : List (Finset (Finset α)))
(hAs : ∀ A ∈ As, UpwardFamily s A) :
fairFamilyMass s (iteratedDisjointOccurrence s As) ≤ (As.map (fairFamilyMass s)).prod := by
induction As with
| nil => simp [iteratedDisjointOccurrence,fairFamilyMass_powerset]
| cons A As ih =>
have hA := hAs A (List.mem_cons_self ..)
have htail : ∀ B ∈ As, UpwardFamily s B := fun B hB => hAs B (List.mem_cons_of_mem _ hB)
exact (finite_fair_BK_mass s A _ hA (upward_iteratedDisjointOccurrence s As)).trans
(by simpa using mul_le_mul_of_nonneg_left (ih htail) (fairFamilyMass_nonneg s A))
lemma mem_foldr_union (ws : List (Finset α)) (a : α) :
a ∈ ws.foldr (· ∪ ·) ∅ ↔ ∃ v ∈ ws, a ∈ v := by
induction ws with
| nil => simp
| cons v vs ih => simp [ih]
theorem certificates_mem_iterated (s : Finset α) (As : List (Finset (Finset α)))
(ws : List (Finset α)) (x : Finset α) (hxs : x ⊆ s)
(hmem : List.Forall₂ (fun w A => w ∈ A) ws As)
(hdis : ws.Pairwise Disjoint) (hsub : ∀ w ∈ ws, w ⊆ x) :
x ∈ iteratedDisjointOccurrence s As := by
induction hmem generalizing x with
| nil => exact mem_powerset.mpr hxs
| @cons w A ws As hw hmem ih =>
have hd := List.pairwise_cons.mp hdis
let u : Finset α := ws.foldr (· ∪ ·) ∅
have hu : ∀ a, a ∈ u ↔ ∃ v ∈ ws, a ∈ v := mem_foldr_union ws
have hux : u ⊆ x := by
intro a ha
obtain ⟨v,hv,hav⟩ := (hu a).mp ha
exact hsub v (List.mem_cons_of_mem _ hv) hav
have htail : ∀ v ∈ ws, v ⊆ u := by
intro v hv a ha
exact (hu a).mpr ⟨v,hv,ha⟩
have hwu : Disjoint w u := by
apply Finset.disjoint_left.mpr
intro a haw hau
obtain ⟨v,hv,hav⟩ := (hu a).mp hau
exact Finset.disjoint_left.mp (hd.1 v hv) haw hav
exact (mem_disjointOccurrenceFamily ..).mpr ⟨hxs,w,hw,u,
ih u (hux.trans hxs) hd.2 htail,hsub w (List.mem_cons_self ..),hux,hwu⟩
noncomputable def certifiedDisjointFamily (s : Finset α) (As : List (Finset (Finset α))) :
Finset (Finset α) := by
classical
exact s.powerset.filter (fun x => ∃ ws : List (Finset α),
List.Forall₂ (fun w A => w ∈ A) ws As ∧ ws.Pairwise Disjoint ∧ ∀ w ∈ ws, w ⊆ x)
theorem finite_fair_BK_certificates (s : Finset α) (As : List (Finset (Finset α)))
(hAs : ∀ A ∈ As, UpwardFamily s A) :
fairFamilyMass s (certifiedDisjointFamily s As) ≤ (As.map (fairFamilyMass s)).prod := by
apply (fairFamilyMass_mono s ?_).trans (finite_fair_BK_repeated s As hAs)
intro x hx
simp only [certifiedDisjointFamily,mem_filter,mem_powerset] at hx
obtain ⟨hxs,ws,hmem,hdis,hsub⟩ := hx
exact certificates_mem_iterated s As ws x hxs hmem hdis hsub
end PlanarFPP
end OAI
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