import Mathlib
import OAI.Computability.VertexCover.Analysis.GraphComplete
import OAI.Computability.VertexCover.Reduction.ReductionSound
import OAI.Computability.VertexCover.Reduction.DenseEncodingBound

namespace OAI

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namespace VertexCover.ClauseProjection
open UniqueGames
noncomputable section

def falseFormula : BinaryFormula.Formula :=
  ⟨[#v[⟨0,true⟩, ⟨0,true⟩, ⟨0,true⟩], #v[⟨0,false⟩, ⟨0,false⟩, ⟨0,false⟩]]⟩

theorem falseFormula_unsat : ¬falseFormula.Satisfiable := by
  rintro ⟨A, hA⟩
  have hpos := hA #v[⟨0,true⟩, ⟨0,true⟩, ⟨0,true⟩] (by simp [falseFormula])
  have hneg := hA #v[⟨0,false⟩, ⟨0,false⟩, ⟨0,false⟩] (by simp [falseFormula])
  cases h : A 0 <;> simp [BinaryFormula.Clause.eval, BinaryFormula.Literal.eval, h] at hpos hneg

def inputFormula (input : List Bool) : BinaryFormula.Formula :=
  (BinaryEncoding.decodeFormula input).getD falseFormula

def InputSAT (input : List Bool) : Prop :=
  ∃ F, BinaryEncoding.decodeFormula input = some F ∧ F.Satisfiable

theorem inputFormula_sat (input : List Bool) : (inputFormula input).Satisfiable ↔ InputSAT input := by
  unfold inputFormula InputSAT
  cases h : BinaryEncoding.decodeFormula input with
  | none => simp [falseFormula_unsat]
  | some F => simp

def binaryReduction (σ : ℝ) (hσ : 0 < σ) (input : List Bool) : VertexCover.LabelCover :=
  reduction σ hσ (BinaryFormula.dense (inputFormula input))

theorem binaryReduction_complete (σ : ℝ) (hσ : 0 < σ) (input : List Bool)
    (h : InputSAT input) :
    ∃ A : (binaryReduction σ hσ input).Labeling,
      ∀ e, (binaryReduction σ hσ input).Satisfies A e :=
  reduction_complete σ hσ _ (BinaryFormula.dense_completeness _ ((inputFormula_sat input).mpr h))

theorem binaryReduction_sound (σ : ℝ) (hσ : 0 < σ) (input : List Bool)
    (h : ¬InputSAT input) : (binaryReduction σ hσ input).value < σ := by
  apply reduction_sound
  intro hs
  exact h ((inputFormula_sat input).mp (BinaryFormula.dense_reflects _ hs))

end
end VertexCover.ClauseProjection

namespace VertexCover
noncomputable section

def rawGraph (m : ℕ) (hm : 4 ≤ m) (input : List Bool) : ExplicitGraph :=
  (ClauseProjection.binaryReduction (Parameters.σ m)
    (Parameters.σ_between m hm).1 input).explicitGraph m

theorem rawGraph_completeness (m : ℕ) (hm : 4 ≤ m) (input : List Bool)
    (h : ThreeSAT input) :
    ((rawGraph m hm input).coverNumber : ℝ) <
      ((1:ℝ)/2+1/(m:ℝ)) * (rawGraph m hm input).n := by
  have hc := ClauseProjection.binaryReduction_complete (Parameters.σ m)
    (Parameters.σ_between m hm).1 input h
  obtain ⟨A, hA⟩ := hc
  exact LabelCover.explicitGraph_complete _ m hm A hA

theorem rawLabelCover_soundness (m : ℕ) (hm : 4 ≤ m) (input : List Bool)
    (h : ¬ ThreeSAT input) :
    (ClauseProjection.binaryReduction (Parameters.σ m)
      (Parameters.σ_between m hm).1 input).value < Parameters.σ m :=
  ClauseProjection.binaryReduction_sound _ _ input h

end
end VertexCover


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end OAI