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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 | import OAI.Combinatorics.ExpectationThreshold.RoundingMain
import OAI.Combinatorics.ExpectationThreshold.ThresholdAttainment
namespace OAI
/-!
# Integral and fractional expectation thresholds
The fractional threshold is genuinely attained with budget one half. Rounding
that attained cover gives an integral cover at the threshold divided by
`25 * 512^4`, still with budget one half and without a support-size restriction.
-/
namespace TalagrandThreshold
variable {α : Type*} [Fintype α] [DecidableEq α]
/-- Talagrand's threshold comparison with the explicit constant `25 * 512^4`.
The coefficient 25 is an integer factor, and both feasible sets use budget
one half. The fractional cover ranges over every subset of the ground set. -/
theorem talagrand_expectation_threshold_equivalence [Nonempty α]
(F : Family α) (_hF : F.Nonempty) (hproper : F ≠ Finset.univ)
(hIncreasing : Increasing F) :
qf F ≤ ((25 : ℝ) * (512 : ℝ) ^ 4) * q F := by
have hempty : (∅ : Finset α) ∉ F := hIncreasing.empty_not_mem hproper
have hqf0 : 0 ≤ qf F := qf_nonneg F hempty
have hqf1 : qf F ≤ 1 := qf_le_one F hempty
have hsmall : Small (qf F / ((25 : ℝ) * (512 : ℝ) ^ 4)) F :=
fractionalSmall_rounding F hqf0 hqf1 (fractionalSmall_qf F hempty)
have hscaled0 : 0 ≤ qf F / ((25 : ℝ) * (512 : ℝ) ^ 4) :=
div_nonneg hqf0 rounding_constant_pos.le
have hscaled1 : qf F / ((25 : ℝ) * (512 : ℝ) ^ 4) ≤ 1 :=
(div_le_one rounding_constant_pos).mpr
(hqf1.trans (by norm_num : (1 : ℝ) ≤ (25 : ℝ) * (512 : ℝ) ^ 4))
have hle := le_q_of_small hscaled0 hscaled1 hsmall
calc
qf F ≤ q F * ((25 : ℝ) * (512 : ℝ) ^ 4) :=
(div_le_iff₀ rounding_constant_pos).mp hle
_ = ((25 : ℝ) * (512 : ℝ) ^ 4) * q F := mul_comm _ _
end TalagrandThreshold
end OAI
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