import Mathlib
import OAI.Computability.VertexCover.Reduction.Soundness
import OAI.Computability.VertexCover.Reduction.GraphSoundness

namespace OAI

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namespace VertexCover.ExplicitGraph

theorem coverNumber_witness (G : ExplicitGraph) :
    ∃ S : Finset (Fin G.n), G.Cover S ∧ S.card = G.coverNumber := by
  have hex : ∃ k : ℕ, ∃ S : Finset (Fin G.n), G.Cover S ∧ S.card = k :=
    ⟨G.n, Finset.univ, by simp [Cover], by simp⟩
  exact Nat.find_spec hex

theorem materialize_cover_reflects {V : Type*} [Fintype V] (G : SimpleGraph V)
    (S : Finset (Fin (materialize G).n)) (hS : (materialize G).Cover S) :
    ∃ C : Finset V, G.IsVertexCover (C : Set V) ∧ C.card = S.card := by
  classical
  let e := Fintype.equivFin V
  let C := S.map e.symm.toEmbedding
  have hmem (v : V) : v ∈ C ↔ e v ∈ S := by
    exact Finset.mem_map_equiv
  refine ⟨C, ?_, Finset.card_map _⟩
  intro v w hvw
  have hne : e v ≠ e w := fun he => G.ne_of_adj hvw (e.injective he)
  rcases lt_or_gt_of_ne hne with hlt | hgt
  · have hp := hS (e v, e w) ((materialize_mem_edges G (e v) (e w)).mpr
      ⟨hlt, by simpa only [e, Equiv.symm_apply_apply] using hvw⟩)
    exact hp.imp (hmem v).mpr (hmem w).mpr
  · have hp := hS (e w, e v) ((materialize_mem_edges G (e w) (e v)).mpr
      ⟨hgt, by simpa only [e, Equiv.symm_apply_apply] using hvw.symm⟩)
    exact hp.symm.imp (hmem v).mpr (hmem w).mpr

theorem materialize_coverNumber_gt {V : Type*} [Fintype V] (G : SimpleGraph V)
    (ρ : ℝ) (h : ∀ C : Finset V, G.IsVertexCover (C : Set V) →
      ρ * Fintype.card V < (C.card:ℝ)) :
    ρ * (materialize G).n < ((materialize G).coverNumber:ℝ) := by
  obtain ⟨S, hS, hmin⟩ := coverNumber_witness (materialize G)
  obtain ⟨C, hC, hcard⟩ := materialize_cover_reflects G S hS
  have hc := h C hC
  rw [hcard, hmin] at hc
  exact hc

end VertexCover.ExplicitGraph

namespace VertexCover.LabelCover

theorem explicitGraph_sound (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
    (hval : Φ.value ≤ Parameters.σ m) :
    (1-1/(m:ℝ))*(Φ.explicitGraph m).n < ((Φ.explicitGraph m).coverNumber:ℝ) := by
  exact ExplicitGraph.materialize_coverNumber_gt _ _ (Φ.graph_soundness_cover m hm hval)

end VertexCover.LabelCover

namespace VertexCover
noncomputable section

theorem rawGraph_soundness (m : ℕ) (hm : 4 ≤ m) (input : List Bool)
    (h : ¬ ThreeSAT input) :
    (1 - 1 / (m : ℝ)) * (rawGraph m hm input).n <
      ((rawGraph m hm input).coverNumber : ℝ) := by
  exact LabelCover.explicitGraph_sound _ m hm
    (le_of_lt (rawLabelCover_soundness m hm input h))

end
end VertexCover
end

section
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end OAI