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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 | 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
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