lean/OAI/Computability/VertexCover/PCP/NameCompaction.lean

import Mathlib
import OAI.Computability.VertexCover.Encoding.Serialization
import OAI.Computability.VertexCover.PCP.ClauseVerifier

namespace OAI

                                                                                       

namespace UniqueGames.Foundations.PCP.NameCompaction

open UniqueGames.Foundations.Target

def clauseNames {n : Nat} («c» : Clause n) : List (Fin n) :=
  [«c»[0].variableIndex, «c»[1].variableIndex, «c»[2].variableIndex]

def sourceNames (F : Formula) : List (Fin F.«variables») :=
  F.clauses.flatMap clauseNames

def activeNames (F : Formula) : List (Fin F.«variables») :=
  (sourceNames F).eraseDups

theorem literal_mem_active (F : Formula) («c» : Clause F.«variables»)
    (hc : «c» ∈ F.clauses) (i : Fin 3) :
    «c»[i].variableIndex ∈ 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 : Fin F.«variables»)
    (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) : Fin F.«variables» :=
  (activeNames F)[v.val]

def compactLiteral (F : Formula) (l : Literal F.«variables»)
    (hl : l.variableIndex ∈ activeNames F) : Literal (activeNames F).length :=
  ⟨compactIndex F l.variableIndex hl, l.positive⟩

def compactClause (F : Formula) («c» : Clause F.«variables»)
    (hc : «c» ∈ F.clauses) : 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 (Clause (activeNames F).length) :=
  F.clauses.attach.map (fun «c» => compactClause F «c».val «c».property)

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

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

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

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

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

@[simp] theorem eval_compact_extend (F : Formula) («c» : Clause F.«variables»)
    (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].variableIndex ∈ activeNames F := literal_mem_active F «c» hc 0
  have h₁ : «c»[1].variableIndex ∈ activeNames F := literal_mem_active F «c» hc 1
  have h₂ : «c»[2].variableIndex ∈ activeNames F := literal_mem_active F «c» hc 2
  simp [compactClause, compactLiteral, Clause.eval, Literal.eval, extendAssignment,
    h₀, h₁, h₂]
  rfl

theorem map_attach_val {α β : Type} (xs : List α) (f : α → β) :
    xs.attach.map (fun x => f x.val) = xs.map f := by
  exact List.attach_map_val

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

def satisfiedCount (F : Formula) (A : Fin F.«variables» → Bool) : Nat :=
  (evaluationList F A).count true

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

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

theorem evaluationList_extend (F : Formula) (B : Fin (compact F).«variables» → Bool) :
    evaluationList (compact F) B = evaluationList F (extendAssignment F B) := by
  simp only [evaluationList, compact, 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
    _ = _ := map_attach_val F.clauses (fun «c» => «c».eval (extendAssignment F B))

theorem satisfiedCount_restrict (F : Formula) (A : Fin F.«variables» → Bool) :
    satisfiedCount (compact F) (restrictAssignment F A) = satisfiedCount F A := by
  simp only [satisfiedCount, evaluationList_restrict]

theorem failedCount_restrict (F : Formula) (A : Fin F.«variables» → Bool) :
    failedCount (compact F) (restrictAssignment F A) = failedCount F A := by
  simp only [failedCount, evaluationList_restrict]

theorem satisfiedCount_extend (F : Formula) (B : Fin (compact F).«variables» → Bool) :
    satisfiedCount (compact F) B = satisfiedCount F (extendAssignment F B) := by
  simp only [satisfiedCount, evaluationList_extend]

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

theorem clauseNames_flat_length {n : Nat} (cs : List (Clause n)) :
    (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 compact_clause_count (F : Formula) :
    (compact F).clauses.length = F.clauses.length := by
  simp [compact, compactClauses]

theorem length_eraseDups_le {α : Type} [BEq α] (xs : List α) :
    xs.eraseDups.length ≤ xs.length := by
  match xs with
  | [] => simp
  | a :: rest =>
    rw [List.eraseDups_cons]
    have filtered := List.length_filter_le (fun b => !(b == a)) rest
    have ih := length_eraseDups_le (rest.filter (fun b => !(b == a)))
    simp only [List.length_cons]
    omega
termination_by xs.length
decreasing_by
  exact Nat.lt_succ_of_le (List.length_filter_le _ _)

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

theorem compact_completeness (F : Formula) (hs : F.Satisfiable) :
    (compact 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 compact_reflects (F : Formula) (hs : (compact F).Satisfiable) :
    F.Satisfiable := by
  rcases hs with ⟨B, hB⟩
  refine ⟨extendAssignment F B, ?_⟩
  intro «c» hc
  have hm : compactClause F «c» hc ∈ (compact 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 compact_satisfiable_iff (F : Formula) :
    (compact F).Satisfiable ↔ F.Satisfiable :=
  ⟨compact_reflects F, compact_completeness F⟩

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

theorem compact_encoding_bound (F : Formula) :
    (Complexity.formulaBits (compact F)).length ≤
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
  have enc := Complexity.formulaBits_length_le (compact F)
  rw [compact_clause_count] at enc
  have active := compact_active_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
    simp [Nat.mul_add, Nat.mul_comm, Nat.mul_left_comm]
    omega
  rw [polynomial] at total
  omega

theorem failureCount_evaluations (F : Formula) (A : Fin F.«variables» → Bool)
    (indices : List (Fin F.clauses.length)) :
    failureCount F A indices =
      (indices.map (fun i => (clauseAt F i).eval A)).count false := by
  induction indices with
  | nil => simp [failureCount]
  | cons i rest ih =>
    cases h : (clauseAt F i).eval A <;>
      simp [failureCount, clauseFailure, h, ih, Nat.add_comm]

theorem evaluations_allIndices (F : Formula) (A : Fin F.«variables» → Bool) :
    ((allIndices F).map (fun i => (clauseAt F i).eval A)) = evaluationList F A := by
  calc
    _ = (List.ofFn (fun i : Fin F.clauses.length => F.clauses[i.val])).map
        (fun «c» => «c».eval A) := by
      simp only [allIndices, List.finRange, clauseAt, List.map_ofFn, Function.comp_def]
    _ = _ := by rw [List.ofFn_getElem]; rfl

theorem verifier_failureCount (F : Formula) (A : Fin F.«variables» → Bool) :
    failureCount F A (allIndices F) = failedCount F A := by
  rw [failureCount_evaluations, evaluations_allIndices]
  rfl

theorem compact_verifier_soundness (F : Formula) (a b : Nat)
    (gap : ∀ A, a * F.clauses.length ≤ b * failedCount F A)
    (alice : AliceStrategy (compact F)) (bob : BobStrategy (compact F)) :
    b * ((allEvents (compact F)).filter (accepts (compact F) alice bob)).length +
        a * F.clauses.length ≤ b * (3 * F.clauses.length) := by
  have sourceGap : ∀ A : BobStrategy (compact F),
      a * (compact F).clauses.length ≤
        b * failureCount (compact F) A (allIndices (compact F)) := by
    intro A
    rw [verifier_failureCount]
    exact compact_preserves_gap F a b gap A
  have h := verifier_event_soundness (compact F) a b sourceGap alice bob
  simpa only [length_allEvents, compact_clause_count] using h

end UniqueGames.Foundations.PCP.NameCompaction

end OAI