import Mathlib
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


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end OAI