import Mathlib
import OAI.Computability.VertexCover.Analysis.DenseSeparatorData
import OAI.Computability.VertexCover.Reduction.RestrictionFrozen

namespace OAI

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namespace VertexCover.Parameters

theorem h_ge_three (m : ℕ) (hm : 0 < m) : 3 ≤ h m := by
  have hpow : 1 ≤ m^3 := one_le_pow₀ hm
  unfold h
  norm_num only [Nat.reducePow]
  omega

theorem private_variance_threshold (m : ℕ) (hm : 0 < m) : ν m*(h m:ℝ)/4 = 8 := by
  have hn : (m:ℝ) ≠ 0 := by positivity
  simp only [ν, h, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
  field_simp
  norm_num

end VertexCover.Parameters

namespace VertexCover.LabelCover
open VertexCover.Restriction

theorem dense_privateVariance_lower (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,
      ∃ J : Finset (Fin (Parameters.d m)), J.card = Parameters.h m ∧
      ∃ frozen : Φ.Seeds (Parameters.d m),
      ∃ weights : Fin (Parameters.d m) → Fin (Φ.WeightDimension (Parameters.d m)) → ℝ,
        Φ.InWeightCube weights ∧
        8 ≤ Φ.ownTotalVariance J frozen (Φ.outsideSum J frozen weights)
          (Φ.highVectors (Parameters.t m) A) hV := by
  obtain ⟨hV, _hzero, henergy⟩ := Φ.dense_separator_data m hm A hA hdense
  rw [← Φ.restrictionFunction_energy_variance] at henergy
  have hdim := Parameters.dimensions m hm
  obtain ⟨J, hJ, frozen, s, hs⟩ := Φ.restriction_frozen_exists
    hdim.1 (Parameters.h_ge_three m (by omega)) hdim.2.2.1.le
    (Parameters.ν m) (Parameters.positive m (by omega)).1
    (Parameters.restriction_budget m hm) _ hV henergy
  refine ⟨hV, J, hJ, frozen, CompactCube.realWeights s, ?_, ?_⟩
  · intro j k
    exact abs_le.mpr (s j k).2
  · simpa only [Parameters.private_variance_threshold m (by omega)] using hs

theorem graph_soundness_independent (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
    (hval : Φ.value ≤ Parameters.σ 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)))) :
    (A.card:ℝ) < (Fintype.card (Φ.Vertex (Parameters.d m)):ℝ)/(m:ℝ) := by
  by_contra hn
  obtain ⟨hV, J, hJ, frozen, weights, _hw, hlow⟩ :=
    Φ.dense_privateVariance_lower m hm A hA (le_of_not_gt hn)
  have hu := Φ.ownTotalVariance_no_upper m hm hval J hJ frozen
    (Φ.outsideSum J frozen weights) _ hV
  linarith

theorem graph_soundness_cover (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
    (hval : Φ.value ≤ Parameters.σ m)
    (C : Finset (Φ.Vertex (Parameters.d m)))
    (hC : (Φ.graph (Parameters.d m) (Parameters.t m)).IsVertexCover
      (C : Set (Φ.Vertex (Parameters.d m)))) :
    (1-1/(m:ℝ)) * Fintype.card (Φ.Vertex (Parameters.d m)) < (C.card:ℝ) := by
  classical
  have hI : (Φ.graph (Parameters.d m) (Parameters.t m)).IsIndepSet
      (↑(Cᶜ) : Set (Φ.Vertex (Parameters.d m))) := by
    simpa only [Finset.coe_compl] using SimpleGraph.isIndepSet_compl_iff_isVertexCover.mpr hC
  have hi := Φ.graph_soundness_independent m hm hval Cᶜ hI
  rw [Finset.card_compl, Nat.cast_sub (Finset.card_le_univ C)] at hi
  simp only [div_eq_mul_inv, one_mul] at hi ⊢
  nlinarith

end VertexCover.LabelCover


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end OAI