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135052a | 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 | import OAI.Probability.SATVariance.LinearLower
import OAI.Probability.SATVariance.ReplacementSharpness
import OAI.Probability.SATVariance.Sharpness
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
theorem variance_main (k : ℕ) (hk : 3 ≤ k) :
(∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, k ≤ n →
variance (H : Stream n k → ℝ) (streamLaw n k) ≤ C * n * ell k n) ∧
(∀ B : ℝ, 0 < B → ∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, k ≤ n →
variance (T B : Stream n k → ℝ) (streamLaw n k) ≤ C * n * ell k n) ∧
(∃ c : ℝ, 0 < c ∧
(∃ N : ℕ, ∀ n : ℕ, N ≤ n → k ≤ n →
c * n ≤ variance (H : Stream n k → ℝ) (streamLaw n k)) ∧
(∀ B : ℝ, U k < B → ∃ N : ℕ, ∀ n : ℕ, N ≤ n → k ≤ n →
c * n ≤ variance (T B : Stream n k → ℝ) (streamLaw n k))) := by
exact ⟨uncapped_variance_upper k hk, capped_variance_upper k hk, variance_linear_lower k hk⟩
end RandomKSAT
end
end OAI
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