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import OAI.Computability.VertexCover.Reduction.HighVectorsSeparated
import OAI.Computability.VertexCover.Reduction.GridBudget
namespace OAI
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namespace VertexCover.Parameters
theorem pair_budget (m : ℕ) (hm : 4 ≤ m) :
(((h m).choose 2 : ℕ) : ℝ)*(ell m : ℝ)^2*σ m = ((h m : ℝ)-1)/(2*(h m : ℝ)) := by
have hpos := positive m (by omega : 0 < m)
have hh : (h m : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hpos.2.2.2.1)
have hl : (ell m : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hpos.2.2.1)
rw [Nat.cast_choose_two, σ_eq]
field_simp
theorem pair_budget_lt (m : ℕ) (hm : 4 ≤ m) :
(((h m).choose 2 : ℕ) : ℝ)*(ell m : ℝ)^2*σ m < (1:ℝ)/2 := by
rw [pair_budget m hm]
have hh : (0:ℝ) < h m := by exact_mod_cast (lt_of_lt_of_le (by decide : 0<2) (dimensions m hm).2.1)
apply (div_lt_iff₀ (by positivity : (0:ℝ) < 2*(h m : ℝ))).mpr
linarith
theorem σ_between (m : ℕ) (hm : 4 ≤ m) : 0 < σ m ∧ σ m < 1 := by
refine ⟨(positive m (by omega)).2.2.2.2.2, ?_⟩
have hm1 : (1:ℝ) ≤ m := by exact_mod_cast (show 1 ≤ m by omega)
have hp : (1:ℝ) ≤ (m:ℝ)^12 := one_le_pow₀ hm1
unfold σ
apply (div_lt_one (by positivity)).mpr
norm_num only [Nat.reducePow]
nlinarith
theorem sqrt_d_gt (m : ℕ) (hm : 4 ≤ m) : 4*(m:ℝ) < Real.sqrt (d m) := by
rw [sqrt_d]
have hm1 : (1:ℝ) ≤ m := by exact_mod_cast (show 1 ≤ m by omega)
have hp : (m:ℝ)^1 ≤ (m:ℝ)^5 := pow_le_pow_right₀ hm1 (by decide)
simp only [pow_one] at hp
norm_num only [Nat.reducePow]
nlinarith
theorem gap_ratio (m : ℕ) (hm : 0 < m) :
(1-1/(m:ℝ))/(1/2+1/(m:ℝ)) = 2-6/((m:ℝ)+2) := by
have hp : (m:ℝ) ≠ 0 := by positivity
have hm2 : (m:ℝ)+2 ≠ 0 := by positivity
have hh : (1:ℝ)/2+1/(m:ℝ) ≠ 0 := by positivity
field_simp
ring
end VertexCover.Parameters
namespace VertexCover.LabelCover
theorem decoding_upper_lt_three_halves (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(hval : Φ.value ≤ Parameters.σ m) (L : Φ.PrivateLists (Parameters.d m))
(hlen : ∀ i, (L i).length ≤ Parameters.ell m)
(J : Finset (Fin (Parameters.d m))) (hJ : J.card = Parameters.h m)
(frozen : Φ.Seeds (Parameters.d m)) :
finiteMean (fun hidden : Φ.HiddenSeeds J =>
(Φ.listHitMaximum L (Φ.spliceSeeds J frozen hidden) J : ℝ)) < 3/2 := by
have hu := Φ.decoding_upper_bound hval L hlen J frozen
rw [hJ] at hu
have hp := Parameters.pair_budget_lt m hm
linarith
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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