File size: 3,013 Bytes
e320e82 | 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 | import OAI.Combinatorics.ExpectationThreshold.SelectorTruncationDefinitions
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.Order.Compact
namespace OAI
/-!
# Existence of the maximal admissible truncation
The condition that every heavy point belongs to `A` is a family of closed
lower bounds on the cutoff, one for each point outside `A`. The excess is a
continuous finite sum. Thus the admissible cutoffs form a closed subset of
`[0, 1]`, and a nonempty such set has a greatest element.
This existence argument does not use positivity or normalization of the
weights, or sign assumptions on the parameters `m` and `d`.
-/
namespace TalagrandThreshold.Selector
open scoped BigOperators
variable {α : Type*} [Fintype α] [DecidableEq α]
/-- Finite truncation gives a continuous excess as the cutoff varies. -/
theorem continuous_truncatedExcess (w : α → ℝ) (A : Finset α) (d : ℝ) :
Continuous (fun u => truncatedExcess w A d u) := by
unfold truncatedExcess
apply continuous_finsetSum
intro x _
exact (continuous_const.min continuous_id).mul continuous_const
/-- The heavy-point inclusion and the excess test are closed conditions. -/
theorem isClosed_admissible (w : α → ℝ) (A : Finset α) (m d : ℝ) :
IsClosed {u : ℝ | Admissible w A m d u} := by
have hlarge : IsClosed {u : ℝ | ∀ x, x ∉ A → w x ≤ u} := by
simp only [Set.ofPred_forall]
exact isClosed_iInter fun x => isClosed_iInter fun _ =>
isClosed_le continuous_const continuous_id
have hexcess : IsClosed {u : ℝ | m * u ≤ truncatedExcess w A d u} :=
isClosed_le (continuous_const.mul continuous_id)
(continuous_truncatedExcess w A d)
have hrepr : {u : ℝ | Admissible w A m d u} =
(Set.Icc 0 1 ∩ {u : ℝ | ∀ x, x ∉ A → w x ≤ u}) ∩
{u : ℝ | m * u ≤ truncatedExcess w A d u} := by
ext u
constructor
· intro hu
exact ⟨⟨⟨hu.1, hu.2.1⟩, fun _ hx => hu.outside_le hx⟩, hu.2.2.2⟩
· intro hu
refine ⟨hu.1.1.1, hu.1.1.2, ?_, hu.2⟩
intro x hux
by_contra hx
exact (not_lt_of_ge (hu.1.2 x hx)) hux
rw [hrepr]
exact (isClosed_Icc.inter hlarge).inter hexcess
/-- Admissible cutoffs are compact because they are closed and lie in `[0, 1]`. -/
theorem isCompact_admissible (w : α → ℝ) (A : Finset α) (m d : ℝ) :
IsCompact {u : ℝ | Admissible w A m d u} := by
apply (isCompact_Icc : IsCompact (Set.Icc (0 : ℝ) 1)).of_isClosed_subset
(isClosed_admissible w A m d)
intro u hu
exact ⟨hu.1, hu.2.1⟩
/-- Every nonempty admissible set has an actual greatest cutoff. -/
theorem exists_greatest_admissible (w : α → ℝ) (A : Finset α) (m d : ℝ)
(hne : ∃ u, Admissible w A m d u) :
∃ ε, Admissible w A m d ε ∧ ∀ u, Admissible w A m d u → u ≤ ε := by
obtain ⟨ε, hε⟩ := (isCompact_admissible w A m d).exists_isGreatest hne
exact ⟨ε, hε.1, fun _ hu => hε.2 hu⟩
end TalagrandThreshold.Selector
end OAI
|