lean/OAI/Computability/VertexCover/Reduction/HonestSelectionCompatible.lean

import Mathlib
import OAI.Computability.VertexCover.PCP.AlphabetTableQueries
import OAI.Computability.VertexCover.Reduction.CompatibleSingleton

namespace OAI

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namespace VertexCover.LabelCover

abbrev PositionPair (d : ℕ) := {jk : Fin d × Fin d // jk.1 < jk.2}
abbrev Seeds (Φ : LabelCover) (d : ℕ) := PositionPair d → Fin Φ.M

def query (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) (j : Fin d) : Φ.Query d :=
  ⟨j, (fun k => Φ.left (seed ⟨(j, k.1), k.2⟩)),
    (fun k => Φ.right (seed ⟨(k.1, j), k.2⟩))⟩

theorem query_left_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (j k : Fin d) (hjk : j < k) :
    Φ.left (seed ⟨(j, k), hjk⟩) ∈ (Φ.query seed j).scopeU := by
  exact Finset.mem_image.mpr ⟨⟨k, hjk⟩, Finset.mem_univ _, rfl⟩

theorem query_right_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (j k : Fin d) (hjk : j < k) :
    Φ.right (seed ⟨(j, k), hjk⟩) ∈ (Φ.query seed k).scopeV := by
  exact Finset.mem_image.mpr ⟨⟨j, hjk⟩, Finset.mem_univ _, rfl⟩

theorem query_injective (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) :
    Function.Injective (Φ.query seed) := by
  intro j k h
  exact congrArg Sigma.fst h

namespace Query

theorem scope_card_le {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) :
    i.scopeU.card + i.scopeV.card ≤ d - 1 := by
  have hU : i.scopeU.card ≤ d - 1 - i.1.val := by
    calc
      i.scopeU.card ≤ Fintype.card {k : Fin d // i.1 < k} := Finset.card_image_le
      _ = (Finset.Ioi i.1).card := by simp [Fintype.card_subtype, Finset.filter_lt_eq_Ioi]
      _ = d - 1 - i.1.val := Fin.card_Ioi i.1
  have hV : i.scopeV.card ≤ i.1.val := by
    calc
      i.scopeV.card ≤ Fintype.card {k : Fin d // k < i.1} := Finset.card_image_le
      _ = (Finset.Iio i.1).card := by simp [Fintype.card_subtype, Finset.filter_gt_eq_Iio]
      _ = i.1.val := Fin.card_Iio i.1
  have := i.1.isLt
  omega

theorem localLabel_card_le {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) :
    Fintype.card i.LocalLabel ≤ (max Φ.qU Φ.qV)^(d - 1) := by
  let q := max Φ.qU Φ.qV
  have hq : 0 < q := lt_of_lt_of_le Φ.qU_pos (le_max_left _ _)
  calc
    Fintype.card i.LocalLabel ≤ Fintype.card i.Assignment := Fintype.card_subtype_le _
    _ = Φ.qU^i.scopeU.card * Φ.qV^i.scopeV.card := by
      simp [Assignment, Fintype.card_prod, ]
    _ ≤ q^i.scopeU.card * q^i.scopeV.card :=
      Nat.mul_le_mul (Nat.pow_le_pow_left (le_max_left _ _) _)
        (Nat.pow_le_pow_left (le_max_right _ _) _)
    _ = q^(i.scopeU.card + i.scopeV.card) := (pow_add _ _ _).symm
    _ ≤ q^(d - 1) := Nat.pow_le_pow_right hq i.scope_card_le

end Query

def localOfSatisfying (Φ : LabelCover) (A : Φ.Labeling)
    (hA : ∀ c, Φ.Satisfies A c) {d : ℕ} (i : Φ.Query d) : i.LocalLabel :=
  ⟨(fun x => A.1 x.1, fun y => A.2 y.1), fun c _ _ => hA c⟩

def honestSelection (Φ : LabelCover) (A : Φ.Labeling)
    (hA : ∀ c, Φ.Satisfies A c) (d : ℕ) : Finset (Φ.Coordinate d) :=
  Finset.univ.image (fun i => ⟨i, Φ.localOfSatisfying A hA i⟩)

theorem honestSelection_compatible (Φ : LabelCover) (A : Φ.Labeling)
    (hA : ∀ c, Φ.Satisfies A c) (d : ℕ) :
    Φ.Compatible (Φ.honestSelection A hA d) := by
  classical
  have shape : ∀ p ∈ Φ.honestSelection A hA d,
      p = ⟨p.1, Φ.localOfSatisfying A hA p.1⟩ := by
    intro p hp
    obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hp
    rfl
  refine ⟨?_, ?_, ?_, ?_⟩
  · intro p hp q hq heq
    calc
      p = ⟨p.1, Φ.localOfSatisfying A hA p.1⟩ := shape p hp
      _ = ⟨q.1, Φ.localOfSatisfying A hA q.1⟩ :=
        congrArg (fun i => (⟨i, Φ.localOfSatisfying A hA i⟩ : Φ.Coordinate d)) heq
      _ = q := (shape q hq).symm
  · intro p hp q hq x hx hy
    have ep := shape p hp
    have eq := shape q hq
    cases p with | mk i a =>
      cases q with | mk j b =>
        simp only [Sigma.mk.inj_iff, heq_eq_eq, true_and] at ep eq
        subst a; subst b; rfl
  · intro p hp q hq y hx hy
    have ep := shape p hp
    have eq := shape q hq
    cases p with | mk i a =>
      cases q with | mk j b =>
        simp only [Sigma.mk.inj_iff, heq_eq_eq, true_and] at ep eq
        subst a; subst b; rfl
  · intro p hp q hq c hx hy
    have ep := shape p hp
    have eq := shape q hq
    cases p with | mk i a =>
      cases q with | mk j b =>
        simp only [Sigma.mk.inj_iff, heq_eq_eq, true_and] at ep eq
        subst a; subst b
        exact hA c

end VertexCover.LabelCover


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end OAI