File size: 2,817 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 77 | import OAI.Combinatorics.ExpectationThreshold.ThresholdCompactness
import OAI.Combinatorics.ExpectationThreshold.ThresholdBounds
namespace OAI
/-!
# Attainment of the integral and fractional thresholds
There are finitely many integral covers, so integral feasibility is closed
in the density interval. Fractional feasibility is a compact projection.
The actual suprema therefore belong to their respective feasible sets as
soon as those sets contain density zero.
-/
namespace TalagrandThreshold
open scoped BigOperators
variable {α : Type*} [Fintype α] [DecidableEq α]
omit [Fintype α] [DecidableEq α] in
theorem continuous_cost (G : Family α) : Continuous (fun p : ℝ => cost p G) := by
unfold cost
exact continuous_finsetSum G (fun S _ => continuous_pow S.card)
omit [DecidableEq α] in
/-- Smallness is a finite union of closed conditions, one per actual cover. -/
theorem isClosed_small (F : Family α) : IsClosed {p : ℝ | Small p F} := by
classical
have hset : {p : ℝ | Small p F} =
⋃ G : Family α, {p : ℝ | Covers G F ∧ cost p G ≤ 1 / 2} := by
ext p
simp only [Set.mem_ofPred_eq, Set.mem_iUnion, Small]
rw [hset]
apply isClosed_iUnion_of_finite
intro G
by_cases hG : Covers G F
· simpa only [hG, true_and] using
(isClosed_le (continuous_cost G) (continuous_const :
Continuous (fun _ : ℝ => (1 / 2 : ℝ))))
· simpa only [hG, false_and, Set.ofPred_false] using
(isClosed_empty : IsClosed (∅ : Set ℝ))
omit [DecidableEq α] in
theorem isCompact_integralFeasible (F : Family α) : IsCompact (integralFeasible F) := by
have hset : integralFeasible F = Set.Icc (0 : ℝ) 1 ∩ {p : ℝ | Small p F} := by
ext p
constructor
· intro hp
exact ⟨⟨hp.1, hp.2.1⟩, hp.2.2⟩
· intro hp
exact ⟨hp.1.1, hp.1.2, hp.2⟩
rw [hset]
exact (isCompact_Icc : IsCompact (Set.Icc (0 : ℝ) 1)).inter_right (isClosed_small F)
omit [DecidableEq α] in
/-- The integral supremum is an actual feasible density. -/
theorem q_mem_integralFeasible (F : Family α) (hF : ∅ ∉ F) :
q F ∈ integralFeasible F := by
exact (isCompact_integralFeasible F).sSup_mem (integralFeasible_nonempty F hF)
/-- The fractional supremum is an actual feasible density. -/
theorem qf_mem_fractionalFeasible (F : Family α) (hF : ∅ ∉ F) :
qf F ∈ fractionalFeasible F := by
exact (isCompact_fractionalFeasible F).sSup_mem (fractionalFeasible_nonempty F hF)
omit [DecidableEq α] in
theorem small_q (F : Family α) (hF : ∅ ∉ F) : Small (q F) F :=
(mem_integralFeasible.mp (q_mem_integralFeasible F hF)).2.2
theorem fractionalSmall_qf (F : Family α) (hF : ∅ ∉ F) : FractionalSmall (qf F) F :=
(mem_fractionalFeasible.mp (qf_mem_fractionalFeasible F hF)).2.2
end TalagrandThreshold
end OAI
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