import Mathlib
import OAI.Computability.VertexCover.Reduction.IncrementNorm

namespace OAI

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namespace VertexCover.LabelCover

def gridRational (d : ℕ) (v : Fin (2*d+1)) : ℚ :=
  2 * ((v.val : ℚ) - d) / (2*d+1)

abbrev Vertex (Φ : LabelCover) (d : ℕ) :=
  Φ.Seeds d × (Fin d → Fin (Φ.WeightDimension d) → Fin (2*d+1))

noncomputable def sumVector (Φ : LabelCover) {d : ℕ} (v : Φ.Vertex d) :
    Φ.Coordinate d → ℝ :=
  ∑ j, Φ.increment (Φ.query v.1 j) (fun k => (gridRational d (v.2 j k) : ℝ))

noncomputable def graph (Φ : LabelCover) (d : ℕ) (t : ℝ) : SimpleGraph (Φ.Vertex d) where
  Adj v w := v ≠ w ∧ Φ.compatibilityNorm (Φ.sumVector v + Φ.sumVector w) ≤ 2*t
  symm := ⟨by intro v w h; exact ⟨h.1.symm, by simpa only [add_comm] using h.2⟩⟩
  loopless := ⟨by intro v h; exact h.1 rfl⟩

theorem low_norm_adjacent (Φ : LabelCover) {d : ℕ} {t : ℝ} {v w : Φ.Vertex d}
    (hne : v ≠ w) (hv : Φ.compatibilityNorm (Φ.sumVector v) ≤ t)
    (hw : Φ.compatibilityNorm (Φ.sumVector w) ≤ t) : (Φ.graph d t).Adj v w := by
  refine ⟨hne, (Φ.norm_add_le _ _).trans ?_⟩
  linarith

theorem low_norm_subsingleton (Φ : LabelCover) {d : ℕ} {t : ℝ}
    (A : Set (Φ.Vertex d)) (hA : (Φ.graph d t).IsIndepSet A) :
    {v ∈ A | Φ.compatibilityNorm (Φ.sumVector v) ≤ t}.Subsingleton := by
  intro v hv w hw
  by_contra hne
  exact ((SimpleGraph.isIndepSet_iff _).mp hA hv.1 hw.1 hne)
    (Φ.low_norm_adjacent hne hv.2 hw.2)

theorem honestSelection_form_increment (Φ : LabelCover) {d : ℕ}
    (A : Φ.Labeling) (hA : ∀ c, Φ.Satisfies A c) (i : Φ.Query d)
    (s : Fin (Φ.WeightDimension d) → ℝ) :
    ∑ p ∈ Φ.honestSelection A hA d, Φ.increment i s p =
      s (i.slot (Φ.localOfSatisfying A hA i)) := by
  classical
  unfold honestSelection
  rw [Finset.sum_image]
  · have hz : ∀ j : Φ.Query d, Φ.increment i s ⟨j, Φ.localOfSatisfying A hA j⟩ =
        if j = i then s (i.slot (Φ.localOfSatisfying A hA i)) else 0 := by
      intro j
      split_ifs with h
      · subst j; exact Φ.increment_same _ _ _
      · exact Φ.increment_other _ _ _ h
    simp only [hz]
    simp
  · intro i _ j _ h
    exact congrArg Sigma.fst h

noncomputable def honestSum (Φ : LabelCover) {d : ℕ} (A : Φ.Labeling)
    (hA : ∀ c, Φ.Satisfies A c) (v : Φ.Vertex d) : ℝ :=
  ∑ j, (gridRational d (v.2 j ((Φ.query v.1 j).slot
    (Φ.localOfSatisfying A hA (Φ.query v.1 j)))) : ℝ)

theorem honestSelection_form_sumVector (Φ : LabelCover) {d : ℕ}
    (A : Φ.Labeling) (hA : ∀ c, Φ.Satisfies A c) (v : Φ.Vertex d) :
    ∑ p ∈ Φ.honestSelection A hA d, Φ.sumVector v p = Φ.honestSum A hA v := by
  classical
  simp only [sumVector, Finset.sum_apply]
  rw [Finset.sum_comm]
  simp only [honestSelection_form_increment, honestSum]

theorem honest_large_sum_independent (Φ : LabelCover) {d : ℕ} (t : ℝ)
    (A : Φ.Labeling) (hA : ∀ c, Φ.Satisfies A c) :
    (Φ.graph d t).IsIndepSet {v | t < Φ.honestSum A hA v} := by
  rw [SimpleGraph.isIndepSet_iff]
  intro v hv w hw _ hadj
  have hform := Φ.form_le_norm (Φ.sumVector v + Φ.sumVector w)
    (Φ.honestSelection A hA d) (Φ.honestSelection_compatible A hA d)
  simp only [Pi.add_apply, Finset.sum_add_distrib, honestSelection_form_sumVector] at hform
  have hsum := (le_abs_self (Φ.honestSum A hA v + Φ.honestSum A hA w)).trans
    (hform.trans hadj.2)
  change t < Φ.honestSum A hA v at hv
  change t < Φ.honestSum A hA w at hw
  linarith

end VertexCover.LabelCover


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end OAI