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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 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 78 79 80 81 82 83 | import OAI.Probability.SATVariance.FailureWindows
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
theorem fixed_cap_variance_lower (k : ℕ) (hk : 3 ≤ k) :
∃ B : ℝ, U k ≤ B ∧ ∃ c : ℝ, 0 < c ∧ ∃ N : ℕ,
∀ n : ℕ, N ≤ n → k ≤ n →
c*n ≤ variance (T B : Stream n k → ℝ) (streamLaw n k) := by
obtain ⟨R,hR,hRU,hRq⟩ := exists_lower_scale k (by omega)
let β := Real.exp (-4*(R:ℝ)*k^2)
have hβ : 0 < β := Real.exp_pos _
have hβ1 : β ≤ 1 := by
apply Real.exp_le_one_iff.mpr
have : (0:ℝ) ≤ R := by positivity
nlinarith [sq_nonneg (k:ℝ)]
obtain ⟨a,ha,ha1,haHall,haβ⟩ := exists_lower_density hβ hβ1
let d := (a/(16*((2*R:ℕ)+1:ℝ)^2))*a*β/16
have hd : 0 < d := by dsimp [d]; positivity
obtain ⟨Ng,hNg⟩ := eventual_geometric_small
(by positivity [real_rate_pos (by omega : 1 ≤ k)] : 0 ≤ 2*(1-((2:ℝ)^k)⁻¹)^R)
hRq (show 0 < β/4 by positivity)
obtain ⟨Na,hNa⟩ := positive_linear_absorb a ha 8
obtain ⟨Nd,hNd⟩ := positive_linear_absorb d hd 4
refine ⟨2*(R:ℝ),hRU,d/4,by positivity,max (2*k^2) (max Ng (max Na Nd)),?_⟩
intro n hn hkn
have hn2 : 2*k^2 ≤ n := by omega
have hng : Ng ≤ n := by omega
have hna : Na ≤ n := by omega
have hnd : Nd ≤ n := by omega
let := clause_nonempty n k hkn
have hs := sentinel_linear_lower (by omega : 2 ≤ k) hkn hR hn2 ha ha1 haHall haβ
(hNa n hna) (hNg n hng)
change d*n ≤ fvariance (fun w : Fin (2*R*n) → Clause n k => (listStop (List.ofFn w):ℝ)) at hs
have hb := sentinel_variance_bridge (n := n) (k := k) (M := 2*R*n)
have hcap : cap n (2*(R:ℝ)) = 2*R*n := by
have he : 2*(R:ℝ)*n = ((2*R*n:ℕ):ℝ) := by
simp only [Nat.cast_mul,Nat.cast_ofNat]
simp only [cap,he,Nat.floor_natCast]
have hv := cap_variance_finite hkn (2*(R:ℝ))
rw [hcap] at hv
rw [←hv] at hb
have hdn := hNd n hnd
nlinarith
theorem variance_linear_lower (k : ℕ) (hk : 3 ≤ k) :
∃ 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
obtain ⟨B₀,hB₀,c₀,hc₀,N₀,hN₀⟩ := fixed_cap_variance_lower k hk
have hk1 : 1 ≤ k := by omega
have hK := bridgeC_nonneg hk1
obtain ⟨N₁,hN₁⟩ := positive_linear_absorb c₀ hc₀ (12*bridgeC k)
refine ⟨c₀/8,by positivity,?_,?_⟩
· refine ⟨max N₀ N₁,fun n hn hkn => ?_⟩
have hl := hN₀ n (by omega) hkn
have ha := hN₁ n (by omega)
have hv := cap_variance_le_H hk1 hkn hB₀
have hn0 : (0:ℝ) ≤ n := by positivity
nlinarith
· intro B hB
refine ⟨max N₀ N₁,fun n hn hkn => ?_⟩
have hl := hN₀ n (by omega) hkn
have ha := hN₁ n (by omega)
have hv := cap_variance_le_H hk1 hkn hB₀
have hv' := H_variance_le_cap hk1 hkn hB.le
nlinarith
end RandomKSAT
end
end OAI
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