File size: 2,862 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 | import OAI.Combinatorics.ExpectationThreshold.Basic
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.Order.Compact
namespace OAI
/-!
# Compactness of the fractional feasible densities
The cover coefficients and the density range over closed unit intervals.
The coverage inequalities and the polynomial cost bound cut out a closed
subset of their compact product. Its projection is exactly the set of
fractionally feasible densities, including when that set is empty.
-/
namespace TalagrandThreshold
noncomputable section
open scoped BigOperators
variable {α : Type*} [Fintype α] [DecidableEq α]
omit [DecidableEq α] in
/-- The fractional cost is jointly continuous in the density and coefficients. -/
theorem continuous_fracCost :
Continuous (fun z : ℝ × (Finset α → ℝ) => fracCost z.1 z.2) := by
unfold fracCost
exact continuous_finsetSum Finset.univ fun S _ =>
((continuous_apply S).comp continuous_snd).mul (continuous_fst.pow S.card)
omit [DecidableEq α] in
/-- Fractional feasibility is the projection of the actual compact set of covers. -/
theorem isCompact_fractionalFeasible (F : Family α) :
IsCompact (fractionalFeasible F) := by
let B : Set (ℝ × (Finset α → ℝ)) :=
Set.Icc (0 : ℝ) 1 ×ˢ {g | ∀ S, 0 ≤ g S ∧ g S ≤ 1}
let C : Set (ℝ × (Finset α → ℝ)) :=
{z | (∀ H ∈ F, 1 ≤ ∑ S ∈ H.powerset, z.2 S) ∧ fracCost z.1 z.2 ≤ 1 / 2}
have hbox : IsCompact B :=
(isCompact_Icc : IsCompact (Set.Icc (0 : ℝ) 1)).prod
(isCompact_pi_infinite fun _ : Finset α =>
(isCompact_Icc : IsCompact (Set.Icc (0 : ℝ) 1)))
have hcoverage : IsClosed
{z : ℝ × (Finset α → ℝ) | ∀ H ∈ F, 1 ≤ ∑ S ∈ H.powerset, z.2 S} := by
simp only [Set.ofPred_forall]
refine isClosed_iInter fun H => isClosed_iInter fun _ => ?_
exact isClosed_le continuous_const
(continuous_finsetSum H.powerset fun S _ =>
(continuous_apply S).comp continuous_snd)
have hcost : IsClosed
{z : ℝ × (Finset α → ℝ) | fracCost z.1 z.2 ≤ 1 / 2} :=
isClosed_le continuous_fracCost continuous_const
have hclosed : IsClosed C := hcoverage.inter hcost
have himage : Prod.fst '' (B ∩ C) = fractionalFeasible F := by
ext p
constructor
· rintro ⟨⟨q, g⟩, ⟨hB, hC⟩, hp⟩
change (0 ≤ q ∧ q ≤ 1) ∧ (∀ S, 0 ≤ g S ∧ g S ≤ 1) at hB
change (∀ H ∈ F, 1 ≤ ∑ S ∈ H.powerset, g S) ∧ fracCost q g ≤ 1 / 2 at hC
change q = p at hp
subst p
exact ⟨hB.1.1, hB.1.2, g, ⟨hB.2, hC.1⟩, hC.2⟩
· rintro ⟨hp0, hp1, g, hcover, hcost⟩
exact ⟨(p, g), ⟨⟨⟨hp0, hp1⟩, hcover.1⟩, ⟨hcover.2, hcost⟩⟩, rfl⟩
rw [← himage]
exact (hbox.inter_right hclosed).image continuous_fst
end
end TalagrandThreshold
end OAI
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