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import OAI.Computability.VertexCover.Analysis.IndependentSecondMomentLower
import OAI.Computability.VertexCover.Analysis.DenseHighFraction
namespace OAI
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namespace VertexCover.Parameters
theorem variance_scale (m : ℕ) (hm : 0 < m) :
b₀ m * ((t m:ℝ)/2)^2 = ν m * (d m:ℝ) := by
have hmR : (m:ℝ) ≠ 0 := by positivity
simp only [b₀, t, ν, d, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
field_simp
end VertexCover.Parameters
namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory
theorem dense_separator_data (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(A : Finset (Φ.Vertex (Parameters.d m)))
(hA : (Φ.graph (Parameters.d m) (Parameters.t m)).IsIndepSet
(A : Set (Φ.Vertex (Parameters.d m))))
(hdense : (Fintype.card (Φ.Vertex (Parameters.d m)):ℝ)/(m:ℝ) ≤ A.card) :
∃ hV : (Φ.highVectors (Parameters.t m) A).Nonempty,
(∀ seed : Φ.Seeds (Parameters.d m),
(∫ s, Φ.separator (Φ.highVectors (Parameters.t m) A) hV (Φ.continuousSum seed s)
∂VertexCover.Cube.blockLaw (Fin (Parameters.d m))
(Fin (Φ.WeightDimension (Parameters.d m)))) = 0) ∧
Parameters.ν m * (Parameters.d m:ℝ) ≤
VertexCover.finiteMean (fun seed : Φ.Seeds (Parameters.d m) =>
variance (fun s => Φ.separator (Φ.highVectors (Parameters.t m) A) hV
(Φ.continuousSum seed s))
(VertexCover.Cube.blockLaw (Fin (Parameters.d m))
(Fin (Φ.WeightDimension (Parameters.d m))))) := by
classical
have hfrac := Φ.dense_high_fraction m hm A hA hdense
have hb0 : 0 < Parameters.b₀ m := by
unfold Parameters.b₀
have hmR : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
positivity
have hBN : (A.filter (fun v => (Parameters.t m:ℝ) <
Φ.compatibilityNorm (Φ.sumVector v))).Nonempty := by
apply Finset.card_pos.mp
by_contra hn
have hz : (A.filter (fun v => (Parameters.t m:ℝ) <
Φ.compatibilityNorm (Φ.sumVector v))).card = 0 := by omega
rw [hz, Nat.cast_zero, zero_div] at hfrac
linarith
have hV : (Φ.highVectors (Parameters.t m) A).Nonempty := hBN.image Φ.sumVector
refine ⟨hV, fun seed => Φ.separator_continuousSum_mean_zero _ hV seed, ?_⟩
have hround : (Parameters.d m:ℝ)/(2*Parameters.d m+1) < (Parameters.t m:ℝ)/2 := by
have h := (Parameters.grid_budget m hm).1.trans (Parameters.grid_budget m hm).2.1
simpa only [Parameters.p, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one] using h
have hlow := Φ.independent_secondMoment_lower (Nat.cast_nonneg (Parameters.t m))
hround A hA hV
have hvareq : ∀ seed : Φ.Seeds (Parameters.d m),
variance (fun s => Φ.separator (Φ.highVectors (Parameters.t m) A) hV
(Φ.continuousSum seed s))
(VertexCover.Cube.blockLaw (Fin (Parameters.d m))
(Fin (Φ.WeightDimension (Parameters.d m)))) =
∫ s, (Φ.separator (Φ.highVectors (Parameters.t m) A) hV
(Φ.continuousSum seed s))^2
∂VertexCover.Cube.blockLaw (Fin (Parameters.d m))
(Fin (Φ.WeightDimension (Parameters.d m))) := by
intro seed
rw [variance_eq_integral
(Φ.separator_continuousSum_continuous _ hV seed).measurable.aemeasurable,
Φ.separator_continuousSum_mean_zero]
simp only [sub_zero]
simp_rw [hvareq]
calc
Parameters.ν m * (Parameters.d m:ℝ) =
Parameters.b₀ m * ((Parameters.t m:ℝ)/2)^2 :=
(Parameters.variance_scale m (by omega)).symm
_ ≤ (((A.filter (fun v => (Parameters.t m:ℝ) <
Φ.compatibilityNorm (Φ.sumVector v))).card:ℝ) /
Fintype.card (Φ.Vertex (Parameters.d m))) * ((Parameters.t m:ℝ)/2)^2 :=
mul_le_mul_of_nonneg_right hfrac (sq_nonneg _)
_ = ((Parameters.t m:ℝ)/2)^2 *
(A.filter (fun v => (Parameters.t m:ℝ) <
Φ.compatibilityNorm (Φ.sumVector v))).card /
Fintype.card (Φ.Vertex (Parameters.d m)) := by ring
_ ≤ _ := hlow
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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