import Mathlib
import OAI.Computability.VertexCover.Reduction.FiniteMeanUnionBound
import OAI.Computability.VertexCover.Reduction.GridBudget

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory

theorem high_count_lower (Φ : LabelCover) {d : ℕ} {t : ℝ}
    (A : Finset (Φ.Vertex d)) (hA : (Φ.graph d t).IsIndepSet (A : Set (Φ.Vertex d))) :
    A.card ≤ (A.filter (fun v => t < Φ.compatibilityNorm (Φ.sumVector v))).card + 1 := by
  classical
  have hl : (A.filter (fun v => Φ.compatibilityNorm (Φ.sumVector v) ≤ t)).card ≤ 1 := by
    apply Finset.card_le_one.mpr
    intro v hv w hw
    exact Φ.low_norm_subsingleton (A : Set (Φ.Vertex d)) hA
      (Finset.mem_filter.mp hv) (Finset.mem_filter.mp hw)
  have he := Finset.card_filter_add_card_filter_not (s := A)
    (fun v => t < Φ.compatibilityNorm (Φ.sumVector v))
  simp only [not_lt] at he
  omega

theorem vertex_card_ge_grid (Φ : LabelCover) {d : ℕ} (hd : 0 < d) :
    2*d+1 ≤ Fintype.card (Φ.Vertex d) := by
  classical
  have hq : 0 < Φ.WeightDimension d := by
    dsimp [WeightDimension]
    exact pow_pos (lt_of_lt_of_le Φ.qU_pos (le_max_left _ _)) _
  have he : 1 ≤ Φ.WeightDimension d * d := Nat.mul_pos hq hd
  rw [Φ.vertex_card]
  have hM : 1 ≤ Φ.M^(d.choose 2) := one_le_pow₀ Φ.M_pos
  have hp : 2*d+1 ≤ (2*d+1)^(Φ.WeightDimension d*d) := by
    simpa using Nat.pow_le_pow_right (by omega : 0 < 2*d+1) he
  exact hp.trans (Nat.le_mul_of_pos_left _ hM)

theorem dense_high_fraction (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) :
    Parameters.b₀ m ≤
      ((A.filter (fun v => (Parameters.t m:ℝ) < Φ.compatibilityNorm (Φ.sumVector v))).card:ℝ) /
        Fintype.card (Φ.Vertex (Parameters.d m)) := by
  classical
  have hmR : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
  have hN : 2*m < Fintype.card (Φ.Vertex (Parameters.d m)) :=
    (Parameters.grid_budget m hm).2.2.trans_le
      (Φ.vertex_card_ge_grid (by have h := (Parameters.dimensions m hm).1; omega))
  have hNR : (2:ℝ)*m < Fintype.card (Φ.Vertex (Parameters.d m)) := by exact_mod_cast hN
  have hNP : (0:ℝ) < Fintype.card (Φ.Vertex (Parameters.d m)) := by linarith
  have hc : (A.card:ℝ) ≤
      (A.filter (fun v => (Parameters.t m:ℝ) < Φ.compatibilityNorm (Φ.sumVector v))).card + 1 := by
    exact_mod_cast Φ.high_count_lower A hA
  unfold Parameters.b₀
  apply (le_div_iff₀ hNP).mpr
  have hd := (div_le_iff₀ hmR).mp hdense
  have hh : (1/(2*(m:ℝ))) * Fintype.card (Φ.Vertex (Parameters.d m)) =
      Fintype.card (Φ.Vertex (Parameters.d m))/(2*(m:ℝ)) := by ring
  rw [hh]
  apply (div_le_iff₀ (by positivity : (0:ℝ)<2*m)).mpr
  nlinarith

end VertexCover.LabelCover

end
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end OAI