File size: 3,220 Bytes
f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Probability.DistinctRoleLaw
import OAI.Combinatorics.BalancedRyser.Selection.LineOccupancyConcentration
namespace OAI
noncomputable section
open Filter Topology
open scoped Classical
namespace Balanced.DistinctRole
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
theorem map_initial_data :
((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).map
(fun M => (initialDeletedLines M, initiallyRetained M)) =
(offsetLaw F).map (fun M => (initialDeletedLines M, initiallyRetained M)) := by
let f : (Direction F → F) → Finset (Line F) × Finset (Point F) :=
fun r => (initialDeletedLines (fun d _ => r d), initiallyRetained (fun d _ => r d))
exact map_role_one_observable f
theorem initial_event_probability (E : Set (Finset (Line F) × Finset (Point F))) :
(((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure
{M | (initialDeletedLines M, initiallyRetained M) ∈ E}).toReal =
offsetEventProbability (F := F)
{M | (initialDeletedLines M, initiallyRetained M) ∈ E} := by
have h := congrArg (fun p : PMF (Finset (Line F) × Finset (Point F)) =>
(p.toOuterMeasure E).toReal) (map_initial_data (F := F))
simp only [PMF.toOuterMeasure_map_apply] at h
convert h using 1 <;> congr 2
theorem initial_failure_probability_eq :
(((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure
{M | ∃ l, l ∉ initialDeletedLines M ∧
((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card F : ℝ) / 5}).toReal =
offsetEventProbability (F := F)
{M | ∃ l, l ∉ initialDeletedLines M ∧
((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card F : ℝ) / 5} := by
exact initial_event_probability
{z | ∃ l, l ∉ z.1 ∧ ((linePoints l ∩ z.2).card : ℝ) < (Fintype.card F : ℝ) / 5}
theorem initial_failure_probability_le
(hgap : (Fintype.card F : ℝ) / 20 ≤
(Fintype.card F : ℝ) * (1 - 1 / (Fintype.card F : ℝ)) ^ Fintype.card F -
(Fintype.card F : ℝ) / 5) :
(((offsetLaw F).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure
{M | ∃ l, l ∉ initialDeletedLines M ∧
((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card F : ℝ) / 5}).toReal ≤
(Fintype.card F : ℝ) * ((Fintype.card F : ℝ) + 1) *
(2 * Real.exp (-(Fintype.card F : ℝ) / 3200)) := by
rw [initial_failure_probability_eq]
exact LineOccupancy.initial_failure_probability_le hgap
theorem tendsto_initial_failure (F : ℕ → Type*)
[∀ n, Field (F n)] [∀ n, Fintype (F n)] [∀ n, DecidableEq (F n)]
(hq : Tendsto (fun n => Fintype.card (F n)) atTop atTop) :
Tendsto (fun n =>
(((offsetLaw (F n)).filter {M | DistinctConstituents M} distinct_support).toOuterMeasure
{M | ∃ l, l ∉ initialDeletedLines M ∧
((linePoints l ∩ initiallyRetained M).card : ℝ) < (Fintype.card (F n) : ℝ) / 5}).toReal)
atTop (𝓝 0) := by
simpa only [initial_failure_probability_eq] using LineOccupancy.tendsto_initial_failure F hq
end Balanced.DistinctRole
end
end OAI
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