import Mathlib
import OAI.Computability.VertexCover.Reduction.NormAttainedSupport
import OAI.Computability.VertexCover.Reduction.PairSeedFiberCard

namespace OAI

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namespace VertexCover.LabelCover

def ListHit (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
    (seed : Φ.Seeds d) (I : Finset (Φ.Coordinate d)) (j : Fin d) : Prop :=
  ∃ a ∈ L (Φ.query seed j), (⟨Φ.query seed j, a⟩ : Φ.Coordinate d) ∈ I

noncomputable def hitPositions (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
    (seed : Φ.Seeds d) (I : Finset (Φ.Coordinate d)) (J : Finset (Fin d)) : Finset (Fin d) := by
  classical
  exact J.filter (Φ.ListHit L seed I)

noncomputable def pairEvents (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
    (seed : Φ.Seeds d) (J : Finset (Fin d)) : Finset (PositionPair d) := by
  classical
  exact Finset.univ.filter (fun e => e.1.1 ∈ J ∧ e.1.2 ∈ J ∧ Φ.PairSeedHit L seed e)

theorem pairSeedHit_of_hits (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
    (seed : Φ.Seeds d) {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I)
    (e : PositionPair d) (hj : Φ.ListHit L seed I e.1.1)
    (hk : Φ.ListHit L seed I e.1.2) : Φ.PairSeedHit L seed e := by
  classical
  obtain ⟨a, haL, haI⟩ := hj
  obtain ⟨b, hbL, hbI⟩ := hk
  refine ⟨a, haL, b, hbL, Φ.compatible_mono hI ?_⟩
  intro p hp
  simp only [Finset.mem_insert, Finset.mem_singleton] at hp
  rcases hp with rfl | rfl
  · exact haI
  · exact hbI

theorem hitPositions_card_le (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
    (seed : Φ.Seeds d) (I : Finset (Φ.Coordinate d)) (hI : Φ.Compatible I)
    (J : Finset (Fin d)) :
    (Φ.hitPositions L seed I J).card ≤ 1 + (Φ.pairEvents L seed J).card := by
  classical
  let A := Φ.hitPositions L seed I J
  by_cases hA : A.Nonempty
  · let j := A.min' hA
    have hj : j ∈ A := Finset.min'_mem _ _
    have keep : ∀ k : ↥(A.erase j), j < k.1 := by
      intro k
      have hk := Finset.mem_erase.mp k.2
      exact lt_of_le_of_ne (Finset.min'_le A k.1 hk.2) (Ne.symm hk.1)
    let inject : ↥(A.erase j) → ↥(Φ.pairEvents L seed J) := fun k =>
      ⟨⟨(j, k.1), keep k⟩, by
        have hmj := Finset.mem_filter.mp hj
        have hmk := Finset.mem_filter.mp (Finset.mem_erase.mp k.2).2
        exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, hmj.1, hmk.1,
          Φ.pairSeedHit_of_hits L seed hI _ hmj.2 hmk.2⟩⟩
    have hinj : Function.Injective inject := by
      intro k l h
      apply Subtype.ext
      exact congrArg (fun e : ↥(Φ.pairEvents L seed J) => e.1.1.2) h
    have hcard := Fintype.card_le_of_injective inject hinj
    simp only [Fintype.card_coe] at hcard
    have herase := Finset.card_erase_add_one hj
    change A.card ≤ _
    omega
  · have he : A = ∅ := Finset.not_nonempty_iff_eq_empty.mp hA
    change A.card ≤ _
    simp [he]

noncomputable def listHitMaximum (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
    (seed : Φ.Seeds d) (J : Finset (Fin d)) : ℕ :=
  (Φ.compatibleSelections d).sup (fun I => (Φ.hitPositions L seed I J).card)

theorem listHitMaximum_le (Φ : LabelCover) {d : ℕ} (L : Φ.PrivateLists d)
    (seed : Φ.Seeds d) (J : Finset (Fin d)) :
    Φ.listHitMaximum L seed J ≤ 1 + (Φ.pairEvents L seed J).card := by
  classical
  apply Finset.sup_le
  intro I hI
  exact Φ.hitPositions_card_le L seed I ((Φ.mem_compatibleSelections I).mp hI) J

end VertexCover.LabelCover


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end OAI