lean/OAI/Computability/VertexCover/Reduction/CompatibleSingleton.lean

import Mathlib
import OAI.Computability.VertexCover.PCP.ZigzagSpectral
import OAI.Computability.VertexCover.PCP.NameCompaction
import OAI.Computability.VertexCover.Repetition.CompletedSampling

namespace OAI

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namespace VertexCover

structure LabelCover where
  u : ℕ
  v : ℕ
  qU : ℕ
  qV : ℕ
  M : ℕ
  qU_pos : 0 < qU
  qV_pos : 0 < qV
  M_pos : 0 < M
  left : Fin M → Fin u
  right : Fin M → Fin v
  projection : Fin M → Fin qU → Fin qV

namespace LabelCover

abbrev Labeling (Φ : LabelCover) := (Fin Φ.u → Fin Φ.qU) × (Fin Φ.v → Fin Φ.qV)

def Satisfies (Φ : LabelCover) (A : Φ.Labeling) (c : Fin Φ.M) : Prop :=
  Φ.projection c (A.1 (Φ.left c)) = A.2 (Φ.right c)

instance (Φ : LabelCover) (A : Φ.Labeling) (c : Fin Φ.M) :
    Decidable (Φ.Satisfies A c) := inferInstanceAs (Decidable (_ = _))

def satisfiedCount (Φ : LabelCover) (A : Φ.Labeling) : ℕ :=
  (Finset.univ.filter (Φ.Satisfies A)).card

def maxSatisfied (Φ : LabelCover) : ℕ :=
  Finset.univ.sup Φ.satisfiedCount

noncomputable def value (Φ : LabelCover) : ℝ := (Φ.maxSatisfied : ℝ) / Φ.M

def defaultLabeling (Φ : LabelCover) : Φ.Labeling :=
  (fun _ => ⟨0, Φ.qU_pos⟩, fun _ => ⟨0, Φ.qV_pos⟩)

theorem satisfiedCount_le_max (Φ : LabelCover) (A : Φ.Labeling) :
    Φ.satisfiedCount A ≤ Φ.maxSatisfied := by
  exact Finset.le_sup (f := Φ.satisfiedCount) (Finset.mem_univ A)

theorem count_le_of_value_le (Φ : LabelCover) {σ : ℝ} (h : Φ.value ≤ σ)
    (A : Φ.Labeling) : (Φ.satisfiedCount A : ℝ) ≤ σ * Φ.M := by
  have hM : (0 : ℝ) < Φ.M := Nat.cast_pos.mpr Φ.M_pos
  have hm : (Φ.maxSatisfied : ℝ) ≤ σ * Φ.M := (div_le_iff₀ hM).mp h
  exact le_trans (Nat.cast_le.mpr (Φ.satisfiedCount_le_max A)) hm

abbrev Query (Φ : LabelCover) (d : ℕ) :=
  Σ j : Fin d,
    ({k : Fin d // j < k} → Fin Φ.u) × ({k : Fin d // k < j} → Fin Φ.v)

namespace Query

def scopeU {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) : Finset (Fin Φ.u) :=
  Finset.univ.image i.2.1

def scopeV {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) : Finset (Fin Φ.v) :=
  Finset.univ.image i.2.2

abbrev Assignment {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) :=
  ({x // x ∈ i.scopeU} → Fin Φ.qU) × ({y // y ∈ i.scopeV} → Fin Φ.qV)

def Valid {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) (A : i.Assignment) : Prop :=
  ∀ c : Fin Φ.M, ∀ hx : Φ.left c ∈ i.scopeU, ∀ hy : Φ.right c ∈ i.scopeV,
    Φ.projection c (A.1 ⟨Φ.left c, hx⟩) = A.2 ⟨Φ.right c, hy⟩

instance {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) (A : i.Assignment) :
    Decidable (i.Valid A) := inferInstanceAs (Decidable (∀ _c _hx _hy, _ = _))

abbrev LocalLabel {Φ : LabelCover} {d : ℕ} (i : Φ.Query d) :=
  {A : i.Assignment // i.Valid A}

end Query

abbrev Coordinate (Φ : LabelCover) (d : ℕ) := Σ i : Φ.Query d, i.LocalLabel

def Compatible (Φ : LabelCover) {d : ℕ} (I : Finset (Φ.Coordinate d)) : Prop :=
  (∀ p ∈ I, ∀ q ∈ I, p.1 = q.1 → p = q) ∧
  (∀ p ∈ I, ∀ q ∈ I, ∀ x (hp : x ∈ p.1.scopeU) (hq : x ∈ q.1.scopeU),
    p.2.1.1 ⟨x, hp⟩ = q.2.1.1 ⟨x, hq⟩) ∧
  (∀ p ∈ I, ∀ q ∈ I, ∀ y (hp : y ∈ p.1.scopeV) (hq : y ∈ q.1.scopeV),
    p.2.1.2 ⟨y, hp⟩ = q.2.1.2 ⟨y, hq⟩) ∧
  (∀ p ∈ I, ∀ q ∈ I, ∀ c (hp : Φ.left c ∈ p.1.scopeU)
      (hq : Φ.right c ∈ q.1.scopeV),
    Φ.projection c (p.2.1.1 ⟨Φ.left c, hp⟩) = q.2.1.2 ⟨Φ.right c, hq⟩)

theorem compatible_mono (Φ : LabelCover) {d : ℕ} {I J : Finset (Φ.Coordinate d)}
    (h : Φ.Compatible I) (hJI : J ⊆ I) : Φ.Compatible J := by
  rcases h with ⟨hu, hx, hy, hc⟩
  exact ⟨fun p hp q hq => hu p (hJI hp) q (hJI hq),
    fun p hp q hq => hx p (hJI hp) q (hJI hq),
    fun p hp q hq => hy p (hJI hp) q (hJI hq),
    fun p hp q hq => hc p (hJI hp) q (hJI hq)⟩

theorem compatible_singleton (Φ : LabelCover) {d : ℕ} (p : Φ.Coordinate d) :
    Φ.Compatible {p} := by
  classical
  refine ⟨?_, ?_, ?_, ?_⟩
  · intro a ha b hb _
    simp only [Finset.mem_singleton] at ha hb
    exact ha.trans hb.symm
  · intro a ha b hb x hx hy
    simp only [Finset.mem_singleton] at ha hb
    subst a; subst b; rfl
  · intro a ha b hb x hx hy
    simp only [Finset.mem_singleton] at ha hb
    subst a; subst b; rfl
  · intro a ha b hb c hx hy
    simp only [Finset.mem_singleton] at ha hb
    subst a; subst b
    exact p.2.2 c hx hy

end LabelCover
end VertexCover


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end OAI