lean/OAI/Computability/MinUncut/Encoding/FormulaEncoding.lean

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

namespace OAI

section
namespace MinUncutGames.Foundations.Complexity

open Target

def parseLiteral («variables» : Nat) : List Nat → Option (Literal «variables» × List Nat)
  | index :: sign :: rest =>
      if bounded : index < «variables» then
        if sign = 0 then some (⟨⟨index, bounded⟩, false⟩, rest)
        else if sign = 1 then some (⟨⟨index, bounded⟩, true⟩, rest)
        else none
      else none
  | _ => none

@[simp] theorem parseLiteral_encoded {n : Nat} (literal : Literal n) (rest : List Nat) :
    parseLiteral n (literalWords literal ++ rest) = some (literal, rest) := by
  cases literal with
  | mk index positive =>
      cases positive <;> simp [literalWords, parseLiteral, index.isLt]

def parseClause («variables» : Nat) (words : List Nat) : Option (Clause «variables» × List Nat) := do
  let (a, words) ← parseLiteral «variables» words
  let (b, words) ← parseLiteral «variables» words
  let (c, words) ← parseLiteral «variables» words
  return (⟨#[a, b, c], rfl⟩, words)

theorem clause_three_entries {n : Nat} (clause : Clause n) :
    (⟨#[clause[0], clause[1], clause[2]], rfl⟩ : Clause n) = clause := by
  apply Vector.ext
  intro i hi
  have cases_i : i = 0 ∨ i = 1 ∨ i = 2 := by omega
  rcases cases_i with rfl | rfl | rfl <;> rfl

@[simp] theorem parseClause_encoded {n : Nat} (clause : Clause n) (rest : List Nat) :
    parseClause n (clauseWords clause ++ rest) = some (clause, rest) := by
  simp [clauseWords, List.append_assoc, parseClause, parseLiteral_encoded,
    clause_three_entries]

def parseClauses («variables» : Nat) : Nat → List Nat → Option (List (Clause «variables») × List Nat)
  | 0, words => some ([], words)
  | count + 1, words => do
      let (clause, words) ← parseClause «variables» words
      let (clauses, words) ← parseClauses «variables» count words
      return (clause :: clauses, words)

@[simp] theorem parseClauses_encoded {n : Nat} (clauses : List (Clause n)) (rest : List Nat) :
    parseClauses n clauses.length (clauses.flatMap clauseWords ++ rest) = some (clauses, rest) := by
  induction clauses with
  | nil => rfl
  | cons clause clauses ih =>
      simp [List.append_assoc, parseClauses, parseClause_encoded, ih]

def decodeFormulaWords : List Nat → Option Formula
  | «variables» :: count :: words => do
      let (clauses, trailing) ← parseClauses «variables» count words
      if trailing = [] then some ⟨«variables», clauses⟩ else none
  | _ => none

@[simp] theorem decodeFormulaWords_encoded (formula : Formula) :
    decodeFormulaWords (formulaWords formula) = some formula := by
  cases formula with
  | mk «variables» clauses =>
      have parsed := parseClauses_encoded clauses []
      simp only [List.append_nil] at parsed
      simp [decodeFormulaWords, formulaWords, parsed]

def decodeFormulaBits (bits : List Bool) : Option Formula :=
  decodeWords bits >>= decodeFormulaWords

@[simp] theorem decodeFormulaBits_encoded (formula : Formula) :
    decodeFormulaBits (formulaBits formula) = some formula := by
  simp [decodeFormulaBits, formulaBits]

theorem formulaBits_injective {first second : Formula}
    (same : formulaBits first = formulaBits second) : first = second := by
  have parsed := congrArg decodeFormulaBits same
  simpa only [decodeFormulaBits_encoded, Option.some.injEq] using parsed

end MinUncutGames.Foundations.Complexity

end
section
namespace MinUncutGames.Foundations.PCP.NameCompaction

open MinUncutGames.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 MinUncutGames.Foundations.PCP.NameCompaction

end

end OAI