lean/OAI/Computability/VertexCover/Reduction/DenseEncodingBound.lean

import Mathlib
import OAI.Computability.VertexCover.Encoding.BinaryFormula
import OAI.Computability.VertexCover.PCP.NameCompaction
import OAI.Computability.VertexCover.Reduction.GridBudget

namespace OAI

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namespace UniqueGames.BinaryFormula
open UniqueGames.Foundations

def activeNames (F : Formula) : List Nat :=
  (sourceNames F).eraseDups

theorem literal_mem_active (F : Formula) («c» : Clause)
    (hc : «c» ∈ F.clauses) (i : Fin 3) : «c»[i].name ∈ activeNames F := by
  simp only [activeNames, List.mem_eraseDups]
  apply List.mem_flatMap.mpr
  refine ⟨«c», hc, ?_⟩
  have hi : i = 0 ∨ i = 1 ∨ i = 2 := by omega
  rcases hi with rfl | rfl | rfl <;> simp [clauseNames]

def compactIndex (F : Formula) (v : Nat)
    (hv : v ∈ activeNames F) : Fin (activeNames F).length :=
  ⟨(activeNames F).idxOf v, List.idxOf_lt_length_iff.mpr hv⟩

def decodeName (F : Formula) (v : Fin (activeNames F).length) : Nat :=
  (activeNames F)[v.val]

def compactLiteral (F : Formula) (l : Literal)
    (hl : l.name ∈ activeNames F) : Target.Literal (activeNames F).length :=
  ⟨compactIndex F l.name hl, l.positive⟩

def compactClause (F : Formula) («c» : Clause)
    (hc : «c» ∈ F.clauses) : Target.Clause (activeNames F).length :=
  #v[compactLiteral F «c»[0] (literal_mem_active F «c» hc 0),
    compactLiteral F «c»[1] (literal_mem_active F «c» hc 1),
    compactLiteral F «c»[2] (literal_mem_active F «c» hc 2)]

def compactClauses (F : Formula) : List (Target.Clause (activeNames F).length) :=
  F.clauses.attach.map (fun «c» => compactClause F «c».val «c».property)

def dense (F : Formula) : Target.Formula where
  «variables» := (activeNames F).length
  clauses := compactClauses F

def restrictAssignment (F : Formula) (A : Nat → Bool) :
    Fin (activeNames F).length → Bool := fun v => A (decodeName F v)

def extendAssignment (F : Formula) (B : Fin (activeNames F).length → Bool)
    (v : Nat) : Bool :=
  if hv : v ∈ activeNames F then B (compactIndex F v hv) else false

@[simp] theorem decode_compactIndex (F : Formula) (v : Nat)
    (hv : v ∈ activeNames F) : decodeName F (compactIndex F v hv) = v := by
  exact List.getElem_idxOf (List.idxOf_lt_length_iff.mpr hv)

theorem compactIndex_injective (F : Formula) (u v : Nat)
    (hu : u ∈ activeNames F) (hv : v ∈ activeNames F)
    (h : compactIndex F u hu = compactIndex F v hv) : u = v := by
  have decoded := congrArg (decodeName F) h
  simpa only [decode_compactIndex] using decoded

@[simp] theorem compactLiteral_positive (F : Formula) (l : Literal)
    (hl : l.name ∈ activeNames F) : (compactLiteral F l hl).positive = l.positive := rfl

@[simp] theorem eval_compact_restrict (F : Formula) («c» : Clause)
    (hc : «c» ∈ F.clauses) (A : Nat → Bool) :
    (compactClause F «c» hc).eval (restrictAssignment F A) = «c».eval A := by
  simp [compactClause, compactLiteral, Target.Clause.eval, Target.Literal.eval,
    Clause.eval, Literal.eval, restrictAssignment, decode_compactIndex]

@[simp] theorem eval_compact_extend (F : Formula) («c» : Clause)
    (hc : «c» ∈ F.clauses) (B : Fin (activeNames F).length → Bool) :
    (compactClause F «c» hc).eval B = «c».eval (extendAssignment F B) := by
  have h₀ : «c»[0].name ∈ activeNames F := literal_mem_active F «c» hc 0
  have h₁ : «c»[1].name ∈ activeNames F := literal_mem_active F «c» hc 1
  have h₂ : «c»[2].name ∈ activeNames F := literal_mem_active F «c» hc 2
  simp [compactClause, compactLiteral, Target.Clause.eval, Target.Literal.eval,
    Clause.eval, Literal.eval, extendAssignment, h₀, h₁, h₂] ; rfl

def evaluationList (F : Formula) (A : Nat → Bool) : List Bool :=
  F.clauses.map (fun «c» => «c».eval A)

theorem evaluationList_restrict (F : Formula) (A : Nat → Bool) :
    PCP.NameCompaction.evaluationList (dense F) (restrictAssignment F A) =
      evaluationList F A := by
  simp only [PCP.NameCompaction.evaluationList, evaluationList, dense, compactClauses,
    List.map_map]
  simpa only [Function.comp_def, eval_compact_restrict] using
    PCP.NameCompaction.map_attach_val F.clauses (fun «c» => «c».eval A)

theorem evaluationList_extend (F : Formula) (B : Fin (dense F).«variables» → Bool) :
    PCP.NameCompaction.evaluationList (dense F) B =
      evaluationList F (extendAssignment F B) := by
  simp only [PCP.NameCompaction.evaluationList, evaluationList, dense, compactClauses,
    List.map_map]
  change F.clauses.attach.map (fun «c» => (compactClause F «c».val «c».property).eval B) = _
  calc
    _ = F.clauses.attach.map (fun «c» => «c».val.eval (extendAssignment F B)) := by
      apply List.map_congr_left
      intro «c» _
      exact eval_compact_extend F «c».val «c».property B
    _ = _ := PCP.NameCompaction.map_attach_val F.clauses
      (fun «c» => «c».eval (extendAssignment F B))

def failedCount (F : Formula) (A : Nat → Bool) : Nat :=
  (evaluationList F A).count false

theorem failedCount_restrict (F : Formula) (A : Nat → Bool) :
    PCP.NameCompaction.failedCount (dense F) (restrictAssignment F A) = failedCount F A := by
  simp only [PCP.NameCompaction.failedCount, failedCount, evaluationList_restrict]

theorem failedCount_extend (F : Formula) (B : Fin (dense F).«variables» → Bool) :
    PCP.NameCompaction.failedCount (dense F) B = failedCount F (extendAssignment F B) := by
  simp only [PCP.NameCompaction.failedCount, failedCount, evaluationList_extend]

theorem clauseNames_flat_length (cs : List Clause) :
    (cs.flatMap clauseNames).length = 3 * cs.length := by
  induction cs with
  | nil => simp
  | cons «c» cs ih =>
    simp only [List.flatMap_cons, List.length_append, clauseNames,
      List.length_cons, List.length_nil, ih]
    omega

@[simp] theorem dense_clause_count (F : Formula) :
    (dense F).clauses.length = F.clauses.length := by
  simp [dense, compactClauses]

theorem dense_variable_bound (F : Formula) :
    (dense F).«variables» ≤ 3 * F.clauses.length := by
  have h := PCP.NameCompaction.length_eraseDups_le (sourceNames F)
  simpa only [sourceNames, clauseNames_flat_length, dense, activeNames] using h

theorem dense_completeness (F : Formula) (hs : F.Satisfiable) :
    (dense F).Satisfiable := by
  rcases hs with ⟨A, hA⟩
  refine ⟨restrictAssignment F A, ?_⟩
  intro d hd
  change d ∈ F.clauses.attach.map (fun «c» => compactClause F «c».val «c».property) at hd
  rcases List.mem_map.mp hd with ⟨«c», _, rfl⟩
  exact (eval_compact_restrict F «c».val «c».property A).trans (hA «c».val «c».property)

theorem dense_reflects (F : Formula) (hs : (dense F).Satisfiable) : F.Satisfiable := by
  rcases hs with ⟨B, hB⟩
  refine ⟨extendAssignment F B, ?_⟩
  intro «c» hc
  have hm : compactClause F «c» hc ∈ (dense F).clauses := by
    change _ ∈ F.clauses.attach.map (fun d => compactClause F d.val d.property)
    exact List.mem_map.mpr ⟨⟨«c», hc⟩, by simp, rfl⟩
  rw [← eval_compact_extend F «c» hc B]
  exact hB _ hm

theorem dense_satisfiable_iff (F : Formula) :
    (dense F).Satisfiable ↔ F.Satisfiable :=
  ⟨dense_reflects F, dense_completeness F⟩

theorem dense_preserves_gap (F : Formula) (a b : Nat)
    (gap : ∀ A, a * F.clauses.length ≤ b * failedCount F A)
    (B : Fin (dense F).«variables» → Bool) :
    a * (dense F).clauses.length ≤ b * PCP.NameCompaction.failedCount (dense F) B := by
  rw [dense_clause_count, failedCount_extend]
  exact gap (extendAssignment F B)

theorem dense_encoding_bound (F : Formula) :
    (Complexity.formulaBits (dense F)).length ≤
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
  have enc := Complexity.formulaBits_length_le (dense F)
  rw [dense_clause_count] at enc
  have active := dense_variable_bound F
  have h₁ := Nat.add_le_add_right active (F.clauses.length + 2)
  have h₂ := Nat.mul_le_mul_left F.clauses.length
    (Nat.mul_le_mul_left 3 (Nat.add_le_add_right active 2))
  have total := Nat.add_le_add h₁ h₂
  have polynomial : 3 * F.clauses.length + (F.clauses.length + 2) +
      F.clauses.length * (3 * (3 * F.clauses.length + 2)) =
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
    ring
  rw [polynomial] at total
  omega

end UniqueGames.BinaryFormula


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end OAI