| import OAI.Probability.FirstPassage.BKRepeated | |
| namespace OAI | |
| namespace PlanarFPP | |
| open Finset | |
| variable {α : Type*} [DecidableEq α] | |
| def ForcesBits (s : Finset α) (A : Finset (Finset α)) (x c : Finset α) : Prop := | |
| ∀ y ⊆ s, (∀ e ∈ c, e ∈ y ↔ e ∈ x) → y ∈ A | |
| lemma ForcesBits.positive_certificate {s : Finset α} {A : Finset (Finset α)} | |
| {x c : Finset α} (hxs : x ⊆ s) (h : ForcesBits s A x c) : x ∩ c ∈ A := by | |
| apply h (x ∩ c) ((inter_subset_left).trans hxs) | |
| intro e he | |
| simp [he] | |
| theorem forcing_certificates_mem (s : Finset α) (As : List (Finset (Finset α))) | |
| (cs : List (Finset α)) (x : Finset α) (hxs : x ⊆ s) | |
| (hforce : List.Forall₂ (fun c A => ForcesBits s A x c) cs As) | |
| (hdis : cs.Pairwise Disjoint) : x ∈ iteratedDisjointOccurrence s As := by | |
| apply certificates_mem_iterated s As (cs.map (fun c => x ∩ c)) x hxs | |
| · clear hdis | |
| induction hforce with | |
| | nil => exact List.Forall₂.nil | |
| | cons h ht ih => exact List.Forall₂.cons (h.positive_certificate hxs) ih | |
| · rw [List.pairwise_map] | |
| exact hdis.imp (fun hab => hab.mono (inter_subset_right) (inter_subset_right)) | |
| · intro w hw | |
| obtain ⟨c,_,rfl⟩ := List.mem_map.mp hw | |
| exact inter_subset_left | |
| theorem finite_fair_BK_forcing (s : Finset α) (As : List (Finset (Finset α))) | |
| (D : Finset (Finset α)) (hAs : ∀ A ∈ As, UpwardFamily s A) | |
| (hD : ∀ x ∈ D, x ⊆ s ∧ ∃ cs : List (Finset α), | |
| List.Forall₂ (fun c A => ForcesBits s A x c) cs As ∧ cs.Pairwise Disjoint) : | |
| fairFamilyMass s D ≤ (As.map (fairFamilyMass s)).prod := by | |
| apply (fairFamilyMass_mono s ?_).trans (finite_fair_BK_repeated s As hAs) | |
| intro x hx | |
| obtain ⟨hxs,cs,hforce,hdis⟩ := hD x hx | |
| exact forcing_certificates_mem s As cs x hxs hforce hdis | |
| end PlanarFPP | |
| end OAI | |