lean/OAI/Computability/DirectedFeedback/Games/InitialGraph.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Encoding.Queries

namespace OAI

namespace DFVSGames.Foundations.PCP.InitialGraph

open Target

abbrev Label := Bool × Bool × Bool
abbrev Vertex (F : Formula) := (Fin F.«variables» ⊕ Fin F.clauses.length) ⊕ Unit
abbrev Dart (F : Formula) := (RandomEvent F × Bool) ⊕ Unit

instance slotFintype : Fintype Slot where
  elems := {Slot.first, Slot.second, Slot.third}
  complete := by intro s; cases s <;> simp

@[simp] theorem card_slot : Fintype.card Slot = 3 := by decide
@[simp] theorem card_label : Fintype.card Label = 8 := by
  simp [Label, Fintype.card_prod]

def variableVertex (F : Formula) (v : Fin F.«variables») : Vertex F := .inl (.inl v)
def clauseVertex (F : Formula) (i : Fin F.clauses.length) : Vertex F := .inl (.inr i)
def dummyVertex (F : Formula) : Vertex F := .inr ()
def dummyDart (F : Formula) : Dart F := .inr ()

instance vertexNonempty (F : Formula) : Nonempty (Vertex F) := ⟨dummyVertex F⟩
instance dartNonempty (F : Formula) : Nonempty (Dart F) := ⟨dummyDart F⟩

def reverse (F : Formula) : Dart F → Dart F
  | .inl (event, orientation) => .inl (event, !orientation)
  | .inr _ => .inr ()

theorem reverse_involutive (F : Formula) : Function.Involutive (reverse F) := by
  intro d
  cases d with
  | inl pair => rcases pair with ⟨e, b⟩; cases b <;> rfl
  | inr u => cases u; rfl

def reverseEquiv (F : Formula) : Dart F ≃ Dart F where
  toFun := reverse F
  invFun := reverse F
  left_inv := reverse_involutive F
  right_inv := reverse_involutive F

def tail (F : Formula) : Dart F → Vertex F
  | .inl (event, false) => clauseVertex F event.1
  | .inl (event, true) => variableVertex F (nameAt (clauseAt F event.1) event.2)
  | .inr _ => dummyVertex F

def toClauseAnswer (label : Label) : ClauseAnswer := ⟨label.1, label.2.1, label.2.2⟩
def fromClauseAnswer (answer : ClauseAnswer) : Label := (answer.first, answer.second, answer.third)
def variableLabel (value : Bool) : Label := (value, false, false)

@[simp] theorem to_fromClauseAnswer (answer : ClauseAnswer) :
    toClauseAnswer (fromClauseAnswer answer) = answer := by
  cases answer; rfl

def variableValid (label : Label) : Bool :=
  decide (label.2.1 = false ∧ label.2.2 = false)

def incidenceAccept (F : Formula) (event : RandomEvent F)
    (clauseLabel variableAnswer : Label) : Bool :=
  variableValid variableAnswer &&
    (localSatisfies (clauseAt F event.1) (toClauseAnswer clauseLabel) &&
      decide (answerAt (toClauseAnswer clauseLabel) event.2 = variableAnswer.1))

def predicate (F : Formula) : Dart F → Label → Label → Bool
  | .inl (event, false), a, b => incidenceAccept F event a b
  | .inl (event, true), a, b => incidenceAccept F event b a
  | .inr _, _, _ => true

def raw (F : Formula) : ConstraintGraph (Vertex F) (Dart F) Label where
  reverse := reverseEquiv F
  reverse_involutive := reverse_involutive F
  tail := tail F
  accepts := predicate F
  reverse_accepts := by
    intro d a b
    cases d with
    | inl pair => rcases pair with ⟨event, orientation⟩; cases orientation <;> rfl
    | inr u => rfl

theorem raw_forward (F : Formula) (labeling : Vertex F → Label) (event : RandomEvent F) :
    (raw F).edgeSatisfied labeling (.inl (event, false)) =
      incidenceAccept F event (labeling (clauseVertex F event.1))
        (labeling (variableVertex F (nameAt (clauseAt F event.1) event.2))) := rfl

theorem incidenceAccept_true (F : Formula) (event : RandomEvent F) (a b : Label)
    (h : incidenceAccept F event a b = true) :
    localSatisfies (clauseAt F event.1) (toClauseAnswer a) = true ∧
      answerAt (toClauseAnswer a) event.2 = b.1 := by
  simp only [incidenceAccept, Bool.and_eq_true, decide_eq_true_eq] at h
  exact h.2

def honestLabeling (F : Formula) (A : Fin F.«variables» → Bool) : Vertex F → Label
  | .inl (.inl v) => variableLabel (A v)
  | .inl (.inr i) => fromClauseAnswer (honestAnswer (clauseAt F i) A)
  | .inr _ => variableLabel false

theorem incidence_honest (F : Formula) (A : Fin F.«variables» → Bool)
    (sat : ∀ c ∈ F.clauses, c.eval A = true) (event : RandomEvent F) :
    incidenceAccept F event (honestLabeling F A (clauseVertex F event.1))
      (honestLabeling F A (variableVertex F (nameAt (clauseAt F event.1) event.2))) = true := by
  have hc : (clauseAt F event.1).eval A = true := sat _ (List.getElem_mem _)
  simp [incidenceAccept, honestLabeling, clauseVertex, variableVertex, variableLabel,
    variableValid, honest_satisfies, honest_answerAt, hc]

theorem raw_completeness (F : Formula) (sat : F.Satisfiable) : (raw F).Satisfiable := by
  rcases sat with ⟨A, hA⟩
  refine ⟨honestLabeling F A, fun d => ?_⟩
  cases d with
  | inl pair =>
    rcases pair with ⟨event, orientation⟩
    cases orientation <;> exact incidence_honest F A hA event
  | inr u => rfl

theorem clauseAnswer_eq_of_slots (a b : ClauseAnswer)
    (h : ∀ s, answerAt a s = answerAt b s) : a = b := by
  rcases a with ⟨a₀, a₁, a₂⟩
  rcases b with ⟨b₀, b₁, b₂⟩
  have h₀ : a₀ = b₀ := h .first
  have h₁ : a₁ = b₁ := h .second
  have h₂ : a₂ = b₂ := h .third
  cases h₀; cases h₁; cases h₂; rfl

theorem raw_reflects (F : Formula) (sat : (raw F).Satisfiable) : F.Satisfiable := by
  rcases sat with ⟨labeling, hlabeling⟩
  let A : Fin F.«variables» → Bool := fun v => (labeling (variableVertex F v)).1
  have slot_correct (i : Fin F.clauses.length) (s : Slot) :
      localSatisfies (clauseAt F i) (toClauseAnswer (labeling (clauseVertex F i))) = true ∧
        answerAt (toClauseAnswer (labeling (clauseVertex F i))) s =
          A (nameAt (clauseAt F i) s) := by
    exact incidenceAccept_true F (i, s) _ _ (hlabeling (.inl ((i, s), false)))
  have clause_correct (i : Fin F.clauses.length) : (clauseAt F i).eval A = true := by
    have equal : toClauseAnswer (labeling (clauseVertex F i)) = honestAnswer (clauseAt F i) A :=
      clauseAnswer_eq_of_slots _ _ (fun s => by
        simpa only [honest_answerAt] using (slot_correct i s).2)
    have satisfied := (slot_correct i .first).1
    rw [equal, honest_satisfies] at satisfied
    exact satisfied
  refine ⟨A, fun c hc => ?_⟩
  obtain ⟨i, hi, heq⟩ := List.getElem_of_mem hc
  simpa only [clauseAt, heq] using clause_correct ⟨i, hi⟩

theorem raw_satisfiable_iff (F : Formula) : (raw F).Satisfiable ↔ F.Satisfiable :=
  ⟨raw_reflects F, raw_completeness F⟩

@[simp] theorem card_vertex (F : Formula) :
    Fintype.card (Vertex F) = F.«variables» + F.clauses.length + 1 := by
  simp [Vertex, Fintype.card_sum]

@[simp] theorem card_dart (F : Formula) :
    Fintype.card (Dart F) = 6 * F.clauses.length + 1 := by
  simp [Dart, RandomEvent, Fintype.card_sum, Fintype.card_prod]
  omega

abbrev CompactVertex (F : Formula) := Vertex (NameCompaction.compact F)
abbrev CompactDart (F : Formula) := Dart (NameCompaction.compact F)

def build (F : Formula) : ConstraintGraph (CompactVertex F) (CompactDart F) Label :=
  raw (NameCompaction.compact F)

theorem build_satisfiable_iff (F : Formula) : (build F).Satisfiable ↔ F.Satisfiable := by
  exact (raw_satisfiable_iff (NameCompaction.compact F)).trans
    (NameCompaction.compact_satisfiable_iff F)

theorem build_vertex_bound (F : Formula) :
    Fintype.card (CompactVertex F) ≤ 4 * F.clauses.length + 1 := by
  have h := NameCompaction.compact_active_bound F
  simp only [CompactVertex, card_vertex, NameCompaction.compact_clause_count]
  omega

theorem build_dart_count (F : Formula) :
    Fintype.card (CompactDart F) = 6 * F.clauses.length + 1 := by
  simp only [CompactDart, card_dart, NameCompaction.compact_clause_count]

theorem build_initial_gap (F : Formula) (unsat : ¬ F.Satisfiable)
    (labeling : CompactVertex F → Label) : 1 ≤ (build F).rejectionCount labeling := by
  apply ConstraintGraph.rejectionCount_positive
  intro h
  exact unsat ((build_satisfiable_iff F).mp h)

theorem build_inverse_size_gap (F : Formula) (unsat : ¬ F.Satisfiable)
    (labeling : CompactVertex F → Label) :
    Fintype.card (CompactDart F) ≤
      (6 * F.clauses.length + 1) * (build F).rejectionCount labeling := by
  rw [build_dart_count]
  simpa only [Nat.mul_one] using
    Nat.mul_le_mul_left (6 * F.clauses.length + 1) (build_initial_gap F unsat labeling)

end DFVSGames.Foundations.PCP.InitialGraph

namespace DFVSGames.Foundations.PCP.AlphabetRetraction

variable {V E A B : Type*}

def pullback (G : ConstraintGraph V E A) (decode : B → A) : ConstraintGraph V E B where
  reverse := G.reverse
  reverse_involutive := G.reverse_involutive
  tail := G.tail
  accepts := fun e b c => G.accepts e (decode b) (decode c)
  reverse_accepts := fun e b c => G.reverse_accepts e (decode b) (decode c)

@[simp] theorem edgeSatisfied_pullback (G : ConstraintGraph V E A)
    (decode : B → A) (labeling : V → B) (e : E) :
    (pullback G decode).edgeSatisfied labeling e =
      G.edgeSatisfied (decode ∘ labeling) e := rfl

@[simp] theorem rejectionCount_pullback [Fintype E] (G : ConstraintGraph V E A)
    (decode : B → A) (labeling : V → B) :
    (pullback G decode).rejectionCount labeling =
      G.rejectionCount (decode ∘ labeling) := rfl

theorem satisfiable_pullback_iff (G : ConstraintGraph V E A)
    (decode : B → A) (encode : A → B) (sectionLaw : ∀ a, decode (encode a) = a) :
    (pullback G decode).Satisfiable ↔ G.Satisfiable := by
  constructor
  · rintro ⟨labeling, h⟩
    exact ⟨decode ∘ labeling, h⟩
  · rintro ⟨labeling, h⟩
    refine ⟨encode ∘ labeling, fun e => ?_⟩
    change G.accepts e (decode (encode (labeling (G.tail e))))
      (decode (encode (labeling (G.head e)))) = true
    simpa only [ConstraintGraph.edgeSatisfied, sectionLaw] using h e

abbrev Label64 := QueryIncidence.Label 6

def decode64 (label : Label64) : InitialGraph.Label := (label 0, label 1, label 2)

def encode64 (label : InitialGraph.Label) : Label64 :=
  fun i => if i = 0 then label.1 else if i = 1 then label.2.1 else label.2.2

@[simp] theorem decode_encode64 (label : InitialGraph.Label) :
    decode64 (encode64 label) = label := by
  rcases label with ⟨a, b, c⟩
  rfl

def initial64 (F : DFVSGames.Foundations.Target.Formula) :
    ConstraintGraph (InitialGraph.CompactVertex F) (InitialGraph.CompactDart F) Label64 :=
  pullback (InitialGraph.build F) decode64

theorem initial64_satisfiable_iff (F : DFVSGames.Foundations.Target.Formula) :
    (initial64 F).Satisfiable ↔ F.Satisfiable :=
  (satisfiable_pullback_iff (InitialGraph.build F) decode64 encode64 decode_encode64).trans
    (InitialGraph.build_satisfiable_iff F)

theorem initial64_alphabet : Fintype.card Label64 = 64 :=
  QueryIncidence.six_query_alphabet

theorem initial64_inverse_size_gap (F : DFVSGames.Foundations.Target.Formula)
    (unsat : ¬ F.Satisfiable) (labeling : InitialGraph.CompactVertex F → Label64) :
    Fintype.card (InitialGraph.CompactDart F) ≤
      (6 * F.clauses.length + 1) * (initial64 F).rejectionCount labeling :=
  InitialGraph.build_inverse_size_gap F unsat (decode64 ∘ labeling)

end DFVSGames.Foundations.PCP.AlphabetRetraction

namespace DFVSGames.Foundations.PCP.RawInitialTables

open Target GraphTables

def unitOrder : Unit ≃ Fin 1 where
  toFun _ := 0
  invFun _ := ()
  left_inv _ := rfl
  right_inv i := by fin_cases i; rfl

def slotOrder : Slot ≃ Fin 3 where
  toFun
    | .first => 0
    | .second => 1
    | .third => 2
  invFun i := if i = 0 then .first else if i = 1 then .second else .third
  left_inv s := by cases s <;> rfl
  right_inv i := by fin_cases i <;> rfl

def binaryOrder (n : Nat) : (Fin n → Bool) ≃ Fin (2 ^ n) where
  toFun bits := (DFVSGames.Integration.BinaryCoordinates.pack bits).toFin
  invFun index := DFVSGames.Integration.BinaryCoordinates.unpack (BitVec.ofFin index)
  left_inv bits := DFVSGames.Integration.BinaryCoordinates.unpack_pack bits
  right_inv index := by
    change (DFVSGames.Integration.BinaryCoordinates.pack
      (DFVSGames.Integration.BinaryCoordinates.unpack (BitVec.ofFin index))).toFin = index
    rw [DFVSGames.Integration.BinaryCoordinates.pack_unpack]

def labelOrder : AlphabetRetraction.Label64 ≃ GraphTables.Label := binaryOrder 6

def vertexOrder (F : Formula) :
    InitialGraph.Vertex F ≃ Fin (F.«variables» + F.clauses.length + 1) :=
  ((finSumFinEquiv.sumCongr unitOrder).trans finSumFinEquiv)

def eventOrder (F : Formula) : RandomEvent F ≃ Fin (F.clauses.length * 3) :=
  ((Equiv.refl (Fin F.clauses.length)).prodCongr slotOrder).trans finProdFinEquiv

def dartOrder (F : Formula) : InitialGraph.Dart F ≃ Fin (6 * F.clauses.length + 1) :=
  (((((eventOrder F).prodCongr finTwoEquiv.symm).trans finProdFinEquiv).sumCongr
    unitOrder).trans finSumFinEquiv).trans (finCongr (by omega))

def raw64 (F : Formula) : ConstraintGraph (InitialGraph.Vertex F)
    (InitialGraph.Dart F) AlphabetRetraction.Label64 :=
  AlphabetRetraction.pullback (InitialGraph.raw F) AlphabetRetraction.decode64

def table (F : Formula) : GraphTables.Table :=
  GraphTables.ofEnumeratedGraph (raw64 F) (vertexOrder F) (dartOrder F) labelOrder

@[simp] theorem table_vertices (F : Formula) :
    (table F).vertices = F.«variables» + F.clauses.length + 1 := rfl

@[simp] theorem table_darts (F : Formula) : (table F).darts = 6 * F.clauses.length + 1 := rfl

theorem table_vertices_positive (F : Formula) : 0 < (table F).vertices := by
  rw [table_vertices]; omega

theorem table_darts_positive (F : Formula) : 0 < (table F).darts := by
  rw [table_darts]; omega

theorem raw64_satisfiable_iff (F : Formula) : (raw64 F).Satisfiable ↔ F.Satisfiable :=
  (AlphabetRetraction.satisfiable_pullback_iff (InitialGraph.raw F)
    AlphabetRetraction.decode64 AlphabetRetraction.encode64
      AlphabetRetraction.decode_encode64).trans (InitialGraph.raw_satisfiable_iff F)

theorem table_semantics (F : Formula) :
    GraphTables.semantics (table F) =
      (raw64 F).reindex (vertexOrder F) (dartOrder F) labelOrder := by
  change GraphTables.semantics (GraphTables.ofGraph
    (GraphTables.enumeratedGraph (raw64 F) (vertexOrder F) (dartOrder F) labelOrder)) = _
  rw [GraphTables.semantics_ofGraph]
  rfl

theorem table_satisfiable_iff (F : Formula) :
    (GraphTables.semantics (table F)).Satisfiable ↔ F.Satisfiable := by
  rw [table_semantics]
  exact ((raw64 F).satisfiable_reindex (vertexOrder F) (dartOrder F) labelOrder).trans
    (raw64_satisfiable_iff F)

theorem initial_rejection (F : Formula) (unsat : ¬ F.Satisfiable)
    (labeling : Fin (table F).vertices → GraphTables.Label) :
    1 ≤ (GraphTables.semantics (table F)).rejectionCount labeling :=
  ConstraintGraph.rejectionCount_positive _
    (fun h => unsat ((table_satisfiable_iff F).mp h)) labeling

theorem initial_size (F : Formula) :
    (table F).vertices + (table F).darts = F.«variables» + 7 * F.clauses.length + 2 := by
  rw [table_vertices, table_darts]
  omega

end DFVSGames.Foundations.PCP.RawInitialTables

noncomputable section

namespace DFVSGames.Foundations.PCP.FiniteGraph

structure Bundle (A : Type) where
  Vertex : Type
  Dart : Type
  vertexFintype : Fintype Vertex
  dartFintype : Fintype Dart
  vertexDecidableEq : DecidableEq Vertex
  dartDecidableEq : DecidableEq Dart
  dartNonempty : Nonempty Dart
  graph : ConstraintGraph Vertex Dart A

namespace Bundle

variable {A : Type}

instance instFintypeVertex (G : Bundle A) : Fintype G.Vertex := G.vertexFintype
instance instFintypeDart (G : Bundle A) : Fintype G.Dart := G.dartFintype
instance instDecidableEqVertex (G : Bundle A) : DecidableEq G.Vertex := G.vertexDecidableEq
instance instDecidableEqDart (G : Bundle A) : DecidableEq G.Dart := G.dartDecidableEq
instance instNonemptyDart (G : Bundle A) : Nonempty G.Dart := G.dartNonempty

instance instNonemptyVertex (G : Bundle A) : Nonempty G.Vertex := by
  obtain ⟨d⟩ := G.dartNonempty
  exact ⟨G.graph.tail d⟩

def ofGraph {V E : Type} [Fintype V] [Fintype E]
    [DecidableEq V] [DecidableEq E] [Nonempty E]
    (G : ConstraintGraph V E A) : Bundle A where
  Vertex := V
  Dart := E
  vertexFintype := inferInstance
  dartFintype := inferInstance
  vertexDecidableEq := inferInstance
  dartDecidableEq := inferInstance
  dartNonempty := inferInstance
  graph := G

def size (G : Bundle A) : Nat := Fintype.card G.Vertex + Fintype.card G.Dart

def Satisfiable (G : Bundle A) : Prop := G.graph.Satisfiable

def rejectionCount (G : Bundle A) (labeling : G.Vertex → A) : Nat :=
  G.graph.rejectionCount labeling

theorem size_positive (G : Bundle A) : 0 < G.size :=
  lt_of_lt_of_le (Fintype.card_pos : 0 < Fintype.card G.Dart) (Nat.le_add_left _ _)

theorem dartCard_le_size (G : Bundle A) : Fintype.card G.Dart ≤ G.size :=
  Nat.le_add_left _ _

@[simp] theorem ofGraph_graph {V E : Type} [Fintype V] [Fintype E]
    [DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
    (ofGraph G).graph = G := rfl

@[simp] theorem ofGraph_size {V E : Type} [Fintype V] [Fintype E]
    [DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
    (ofGraph G).size = Fintype.card V + Fintype.card E := rfl

@[simp] theorem ofGraph_satisfiable {V E : Type} [Fintype V] [Fintype E]
    [DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
    (ofGraph G).Satisfiable ↔ G.Satisfiable := Iff.rfl

@[simp] theorem ofGraph_rejectionCount {V E : Type} [Fintype V] [Fintype E]
    [DecidableEq V] [DecidableEq E] [Nonempty E]
    (G : ConstraintGraph V E A) (labeling : V → A) :
    (ofGraph G).rejectionCount labeling = G.rejectionCount labeling := rfl

variable [Fintype A] [Nonempty A]

def minimumRejections (G : Bundle A) : Nat := G.graph.minimumRejections

def gap (G : Bundle A) : ℝ := G.graph.gap

theorem exists_minimizer (G : Bundle A) :
    ∃ labeling : G.Vertex → A, G.rejectionCount labeling = G.minimumRejections :=
  G.graph.exists_minimizer

theorem gap_nonnegative (G : Bundle A) : 0 ≤ G.gap := G.graph.gap_nonnegative

theorem gap_le_one (G : Bundle A) : G.gap ≤ 1 := G.graph.gap_le_one

theorem gap_eq_zero_iff (G : Bundle A) : G.gap = 0 ↔ G.Satisfiable :=
  G.graph.gap_eq_zero_iff

theorem le_gap_iff (G : Bundle A) (ε : ℝ) :
    ε ≤ G.gap ↔ ∀ labeling : G.Vertex → A,
      ε * Fintype.card G.Dart ≤ (G.rejectionCount labeling : ℝ) :=
  G.graph.le_gap_iff ε

theorem inverse_card_le_gap_of_unsatisfiable (G : Bundle A) (unsat : ¬ G.Satisfiable) :
    1 / (Fintype.card G.Dart : ℝ) ≤ G.gap :=
  G.graph.inverse_card_le_gap_of_unsatisfiable unsat

theorem one_le_dartCard_mul_gap (G : Bundle A) (unsat : ¬ G.Satisfiable) :
    1 ≤ (Fintype.card G.Dart : ℝ) * G.gap := by
  have he : (0 : ℝ) < Fintype.card G.Dart := by exact_mod_cast Fintype.card_pos
  have h := (div_le_iff₀ he).mp (G.inverse_card_le_gap_of_unsatisfiable unsat)
  simpa only [mul_comm] using h

theorem one_le_size_mul_gap (G : Bundle A) (unsat : ¬ G.Satisfiable) :
    1 ≤ (G.size : ℝ) * G.gap := by
  have hsize : (Fintype.card G.Dart : ℝ) ≤ (G.size : ℝ) := by
    exact_mod_cast G.dartCard_le_size
  exact (G.one_le_dartCard_mul_gap unsat).trans
    (mul_le_mul_of_nonneg_right hsize G.gap_nonnegative)

@[simp] theorem ofGraph_minimumRejections {V E : Type} [Fintype V] [Fintype E]
    [DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
    (ofGraph G).minimumRejections = G.minimumRejections := rfl

@[simp] theorem ofGraph_gap {V E : Type} [Fintype V] [Fintype E]
    [DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
    (ofGraph G).gap = G.gap := rfl

end Bundle

end DFVSGames.Foundations.PCP.FiniteGraph

end

namespace DFVSGames.Foundations.PCP.FinalConstants

def alphabet : Nat := 64
def compositionLoss : Nat := 12288
def denominator : Nat := 16 * alphabet ^ 4 * 66 * compositionLoss
def windowHalf : Nat := denominator * Preprocessing.sizeFactor
def windowSize : Nat := 2 * windowHalf + 1
def smoothingScale : Nat := 4 * alphabet * windowHalf
def endpointLength : Nat := smoothingScale ^ 2
def walkLength : Nat := 2 * endpointLength + 1

noncomputable def gain : ℝ := (windowSize : ℝ) / denominator
noncomputable def cap : ℝ := 1 / (walkLength : ℝ)

theorem denominator_positive : 0 < denominator := by
  norm_num [denominator, alphabet, compositionLoss]

theorem windowHalf_positive : 0 < windowHalf :=
  Nat.mul_pos denominator_positive Preprocessing.sizeFactor_positive

theorem walkLength_positive : 0 < walkLength := by
  simp [walkLength]

theorem gain_large : 2 * (Preprocessing.sizeFactor : ℝ) ≤ gain := by
  have hd : (0 : ℝ) < denominator := by exact_mod_cast denominator_positive
  apply (le_div_iff₀ hd).2
  simp only [windowSize, windowHalf, Nat.cast_add, Nat.cast_mul,
    Nat.cast_ofNat]
  nlinarith

theorem cap_positive : 0 < cap := by
  exact one_div_pos.mpr (by exact_mod_cast walkLength_positive)

theorem cap_le_one : cap ≤ 1 := by
  apply (div_le_one (by exact_mod_cast walkLength_positive)).2
  exact_mod_cast walkLength_positive

theorem composed_gap (epsilon : ℝ) (he : 0 ≤ epsilon) :
    min (2 * epsilon) cap ≤
      gain * min (epsilon / (Preprocessing.sizeFactor : ℝ))
        (1 / (walkLength : ℝ)) := by
  have hs : (0 : ℝ) < Preprocessing.sizeFactor := by
    exact_mod_cast Preprocessing.sizeFactor_positive
  have hs1 : (1 : ℝ) ≤ Preprocessing.sizeFactor := by
    exact_mod_cast Preprocessing.sizeFactor_positive
  have hg : 0 ≤ gain := le_trans (by positivity) gain_large
  rw [mul_min_of_nonneg _ _ hg]
  apply min_le_min
  · calc
      2 * epsilon = (2 * (Preprocessing.sizeFactor : ℝ)) *
          (epsilon / (Preprocessing.sizeFactor : ℝ)) := by field_simp
      _ ≤ gain * (epsilon / (Preprocessing.sizeFactor : ℝ)) :=
        mul_le_mul_of_nonneg_right gain_large (div_nonneg he hs.le)
  · change cap ≤ gain * cap
    have hgain : 1 ≤ gain := by linarith [gain_large]
    simpa only [one_mul] using mul_le_mul_of_nonneg_right hgain cap_positive.le

end DFVSGames.Foundations.PCP.FinalConstants

namespace DFVSGames.Foundations.PCP.RoundTables

abbrev BaseTable := PreprocessingTables.BaseTable

opaque fixedWalkParameter : {n : Nat // n = 2 * FinalConstants.endpointLength} :=
  ⟨2 * FinalConstants.endpointLength, rfl⟩

def walkParameter : Nat := fixedWalkParameter.val

theorem walkParameter_eq : walkParameter = 2 * FinalConstants.endpointLength :=
  fixedWalkParameter.property

def alphabet : Nat := PoweringTables.labelCount PreprocessingTables.degree walkParameter

theorem alphabet_positive : 0 < alphabet := by
  unfold alphabet PoweringTables.labelCount
  exact Nat.pow_pos (by decide)

def powered (H : BaseTable) (table : GraphTables.Table) : GenericGraphTables.Table alphabet :=
  PoweringTables.table (PreprocessingTables.preprocess H table) walkParameter

def build (H : BaseTable) (table : GraphTables.Table) : GraphTables.Table :=
  AlphabetTable.Table.build (powered H table)

def sizeFactor : Nat := AlphabetGraphBounds.sizeFactor alphabet *
  (ExpanderFamily.growth ^ 2 * (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1)))

theorem degree_positive : 0 < PreprocessingTables.degree := by
  unfold PreprocessingTables.degree
  omega

theorem dartFactor_positive (q : Nat) : 0 < AlphabetGraphBounds.dartFactor q := by
  unfold AlphabetGraphBounds.dartFactor AlphabetGraphBounds.localEventFactor
  positivity

theorem sizeFactor_positive : 0 < sizeFactor := by
  unfold sizeFactor
  apply Nat.mul_pos (AlphabetGraphBounds.sizeFactor_positive alphabet)
  apply Nat.mul_pos
  · exact Nat.pow_pos (Nat.zero_lt_one.trans ExpanderFamily.growth_gt_one)
  · omega

theorem powered_vertices (H : BaseTable) (table : GraphTables.Table) :
    (powered H table).vertices = PreprocessingTables.vertices table := rfl

theorem powered_darts (H : BaseTable) (table : GraphTables.Table) :
    (powered H table).darts = 2 * PreprocessingTables.vertices table *
      PreprocessingTables.degree ^ (walkParameter + 1) := rfl

theorem build_darts_positive (H : BaseTable) (table : GraphTables.Table) :
    0 < (build H table).darts := by
  change 0 < (AlphabetTable.Table.build (powered H table)).darts
  rw [AlphabetTableBounds.build_darts, powered_darts]
  exact Nat.mul_pos (Nat.mul_pos
    (Nat.mul_pos (by decide) (PreprocessingTables.vertices_positive table))
    (Nat.pow_pos degree_positive)) (dartFactor_positive alphabet)

theorem build_size_le (H : BaseTable) (table : GraphTables.Table)
    (ht : 0 < table.darts) :
    (build H table).vertices + (build H table).darts ≤
      sizeFactor * (table.vertices + table.darts) := by
  have hv := PreprocessingTables.vertices_le_of_positive table ht
  have he : (powered H table).vertices + (powered H table).darts =
      PreprocessingTables.vertices table *
        (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1)) := by
    rw [powered_vertices, powered_darts]
    ring
  have hp := Nat.mul_le_mul_right
    (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1)) hv
  calc
    _ ≤ AlphabetGraphBounds.sizeFactor alphabet *
        ((powered H table).vertices + (powered H table).darts) :=
      AlphabetTableBounds.build_total_le (powered H table)
    _ = AlphabetGraphBounds.sizeFactor alphabet *
        (PreprocessingTables.vertices table *
          (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1))) := by rw [he]
    _ ≤ AlphabetGraphBounds.sizeFactor alphabet *
        ((ExpanderFamily.growth ^ 2 * table.darts) *
          (1 + 2 * PreprocessingTables.degree ^ (walkParameter + 1))) :=
      Nat.mul_le_mul_left _ hp
    _ = sizeFactor * table.darts := by unfold sizeFactor; ring
    _ ≤ sizeFactor * (table.vertices + table.darts) :=
      Nat.mul_le_mul_left _ (Nat.le_add_left _ _)

theorem build_completeness (H : BaseTable) (table : GraphTables.Table)
    (sat : (GraphTables.semantics table).Satisfiable) :
    (GraphTables.semantics (build H table)).Satisfiable := by
  let : Nonempty (Fin alphabet) := ⟨⟨0, alphabet_positive⟩⟩
  exact AlphabetTable.Table.perfect_completeness (powered H table)
    (PoweringTableSemantics.preserves_satisfiability
      (PreprocessingTables.preprocess H table) walkParameter
      (PreprocessingGuarantees.completeness H table sat))

abbrev Input := {table : GraphTables.Table // 0 < table.darts}

instance (input : Input) : Nonempty (Fin input.val.darts) := ⟨⟨0, input.property⟩⟩

def step (H : BaseTable) (input : Input) : Input :=
  ⟨build H input.val, build_darts_positive H input.val⟩

def initial (F : Target.Formula) : Input :=
  ⟨RawInitialTables.table F, RawInitialTables.table_darts_positive F⟩

def size (input : Input) : Nat := input.val.vertices + input.val.darts

def Satisfiable (input : Input) : Prop := (GraphTables.semantics input.val).Satisfiable

noncomputable def gap (input : Input) : ℝ := (GraphTables.semantics input.val).gap

theorem size_positive (input : Input) : 0 < size input :=
  input.property.trans_le (Nat.le_add_left _ _)

theorem gap_nonnegative (input : Input) : 0 ≤ gap input :=
  (GraphTables.semantics input.val).gap_nonnegative

theorem gap_eq_zero_iff (input : Input) : gap input = 0 ↔ Satisfiable input :=
  (GraphTables.semantics input.val).gap_eq_zero_iff

theorem one_le_size_mul_gap (input : Input) (unsat : ¬ Satisfiable input) :
    1 ≤ (size input : ℝ) * gap input := by
  have h := (GraphTables.semantics input.val).inverse_card_le_gap_of_unsatisfiable unsat
  have hp : (0 : ℝ) < input.val.darts := Nat.cast_pos.mpr input.property
  simp only [Fintype.card_fin] at h
  have hd := (div_le_iff₀ hp).mp h
  have hsize : (input.val.darts : ℝ) ≤ (size input : ℝ) := by
    exact_mod_cast (Nat.le_add_left input.val.darts input.val.vertices)
  exact (show 1 ≤ (input.val.darts : ℝ) * gap input by
    simpa only [gap, mul_comm] using hd).trans
    (mul_le_mul_of_nonneg_right hsize (gap_nonnegative input))

theorem initial_satisfiable_iff (F : Target.Formula) : Satisfiable (initial F) ↔ F.Satisfiable :=
  RawInitialTables.table_satisfiable_iff F

theorem step_completeness (H : BaseTable) (input : Input) :
    Satisfiable input → Satisfiable (step H input) := build_completeness H input.val

theorem step_size (H : BaseTable) (input : Input) :
    size (step H input) ≤ sizeFactor * size input := build_size_le H input.val input.property

end DFVSGames.Foundations.PCP.RoundTables

namespace DFVSGames.Foundations.PCP.AmplificationIteration

def run {X : Type*} (step : X → X) : Nat → X → X
  | 0, x => x
  | n + 1, x => step (run step n x)

private theorem min_double_min_inline_AmplificationIteration (u cap : ℝ) (hc : 0 ≤ cap) :
    min (2 * min u cap) cap = min (2 * u) cap := by
  by_cases hu : u ≤ cap
  · rw [min_eq_left hu]
  · have hcu : cap ≤ u := le_of_not_ge hu
    rw [min_eq_right hcu, min_eq_right (by linarith : cap ≤ 2 * cap),
      min_eq_right (by linarith : cap ≤ 2 * u)]

theorem run_gap {X : Type*} (step : X → X) (gap : X → ℝ)
    (cap : ℝ) (hc : 0 ≤ cap)
    (amplifies : ∀ x, min (2 * gap x) cap ≤ gap (step x))
    (n : Nat) (x : X) :
    min (2 ^ n * gap x) cap ≤ gap (run step n x) := by
  induction n with
  | zero => simp [run]
  | succ n ih =>
    have hm : min (2 * min (2 ^ n * gap x) cap) cap ≤
        min (2 * gap (run step n x)) cap :=
      min_le_min (mul_le_mul_of_nonneg_left ih (by norm_num)) le_rfl
    rw [min_double_min_inline_AmplificationIteration _ _ hc] at hm
    have h := hm.trans (amplifies (run step n x))
    simpa [run, pow_succ, mul_assoc, mul_left_comm, mul_comm] using h

theorem run_preserves {X : Type*} (step : X → X) (P : X → Prop)
    (preserves : ∀ x, P x → P (step x)) (n : Nat) (x : X) (hx : P x) :
    P (run step n x) := by
  induction n with
  | zero => exact hx
  | succ n ih => exact preserves _ ih

theorem run_size {X : Type*} (step : X → X) (size : X → Nat) (C : Nat)
    (blowup : ∀ x, size (step x) ≤ C * size x) (n : Nat) (x : X) :
    size (run step n x) ≤ C ^ n * size x := by
  induction n with
  | zero => simp [run]
  | succ n ih =>
    have h := (blowup (run step n x)).trans (Nat.mul_le_mul_left C ih)
    simpa [run, pow_succ, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm] using h

def rounds (n : Nat) : Nat := Nat.log2 n + 1

theorem rounds_large (n : Nat) : n < 2 ^ rounds n := by
  simpa [rounds, Nat.log2_eq_log_two] using
    (Nat.lt_pow_succ_log_self (by decide : 1 < 2) n)

theorem rounds_power_le {n : Nat} (hn : 0 < n) : 2 ^ rounds n ≤ 2 * n := by
  have h := Nat.pow_log_le_self 2 (Nat.ne_of_gt hn)
  simpa [rounds, Nat.log2_eq_log_two, pow_succ, Nat.mul_comm] using
    (Nat.mul_le_mul_right 2 h)

theorem blowup_rounds_le {C degree n : Nat} (hn : 0 < n) (hC : C ≤ 2 ^ degree) :
    C ^ rounds n ≤ (2 * n) ^ degree := by
  calc
    C ^ rounds n ≤ (2 ^ degree) ^ rounds n := Nat.pow_le_pow_left hC _
    _ = (2 ^ rounds n) ^ degree := by
      rw [← pow_mul, ← pow_mul, Nat.mul_comm degree (rounds n)]
    _ ≤ (2 * n) ^ degree := Nat.pow_le_pow_left (rounds_power_le hn) _

theorem run_size_polynomial {X : Type*} (step : X → X) (size : X → Nat)
    {C degree n : Nat} (hn : 0 < n) (hC : C ≤ 2 ^ degree)
    (blowup : ∀ x, size (step x) ≤ C * size x) (x : X) :
    size (run step (rounds n) x) ≤ (2 * n) ^ degree * size x :=
  (run_size step size C blowup (rounds n) x).trans
    (Nat.mul_le_mul_right _ (blowup_rounds_le hn hC))

theorem run_reaches_cap {X : Type*} (step : X → X) (gap : X → ℝ)
    (cap : ℝ) (hc : 0 ≤ cap) (hc1 : cap ≤ 1)
    (amplifies : ∀ x, min (2 * gap x) cap ≤ gap (step x))
    (n : Nat) (x : X) (hg : 0 ≤ gap x) (hstart : 1 ≤ (n : ℝ) * gap x) :
    cap ≤ gap (run step (rounds n) x) := by
  have hp : (n : ℝ) ≤ (2 : ℝ) ^ rounds n := by
    exact_mod_cast (Nat.le_of_lt (rounds_large n))
  have hcap : cap ≤ (2 : ℝ) ^ rounds n * gap x :=
    hc1.trans (hstart.trans (mul_le_mul_of_nonneg_right hp hg))
  have h := run_gap step gap cap hc amplifies (rounds n) x
  rwa [min_eq_right hcap] at h

end DFVSGames.Foundations.PCP.AmplificationIteration

end OAI