lean/OAI/Computability/DirectedFeedback/RankGraph/CloneGap.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.ContextLoad

namespace OAI


namespace DFVSGames.Reduction.CloneGap

variable {Element OtherElement : Type*}

def cart (xs : List Element) (ys : List OtherElement) : List (Element × OtherElement) :=
  xs.flatMap fun x => ys.map fun y => (x, y)

theorem count_const (xs : List Element) (b : Bool) :
    xs.countP (fun _ => b) = if b then xs.length else 0 := by
  cases b <;> simp

theorem count_cart (xs : List Element) (ys : List OtherElement)
    (p : Element → Bool) (q : OtherElement → Bool) :
    (cart xs ys).countP (fun z => p z.1 && q z.2) =
      xs.countP p * ys.countP q := by
  induction xs with
  | nil => simp [cart]
  | cons x xs ih =>
      simp only [cart, List.flatMap_cons, List.countP_append] at *
      rw [ih]
      cases h : p x <;>
        simp [List.countP_map, Function.comp_def, h, Nat.add_mul, Nat.add_comm]

theorem length_cart (xs : List Element) (ys : List OtherElement) :
    (cart xs ys).length = xs.length * ys.length := by
  have := count_cart xs ys (fun _ => true) (fun _ => true)
  simpa using this

theorem count_cart_left (xs : List Element) (ys : List OtherElement) (p : Element → Bool) :
    (cart xs ys).countP (fun z => p z.1) = xs.countP p * ys.length := by
  simpa using count_cart xs ys p (fun _ => true)

theorem count_cart_right (xs : List Element) (ys : List OtherElement) (p : OtherElement → Bool) :
    (cart xs ys).countP (fun z => p z.2) = xs.length * ys.countP p := by
  simpa using count_cart xs ys (fun _ => true) p

theorem count_cart_le (xs : List Element) (ys : List OtherElement)
    (p : Element × OtherElement → Bool)
    (bound : Nat) (h : ∀ x ∈ xs, ys.countP (fun y => p (x, y)) ≤ bound) :
    (cart xs ys).countP p ≤ xs.length * bound := by
  induction xs with
  | nil => simp [cart]
  | cons x xs ih =>
      have hx := h x (by simp)
      have ht := ih (fun y hy => h y (by simp [hy]))
      simp only [cart, List.flatMap_cons, List.countP_append, List.countP_map,
        Function.comp_def] at *
      simp only [List.length_cons, Nat.add_mul]
      omega

theorem count_union_le (xs : List Element) (p q : Element → Bool) :
    xs.countP (fun x => p x || q x) ≤ xs.countP p + xs.countP q := by
  induction xs with
  | nil => simp
  | cons x xs ih =>
      cases hp : p x <;> cases hq : q x <;>
        simp [hp, hq] at * <;> omega

theorem count_remove (xs : List Element) (p q : Element → Bool) :
    xs.countP p ≤ xs.countP (fun x => p x && !q x) + xs.countP q := by
  induction xs with
  | nil => simp
  | cons x xs ih =>
      cases hp : p x <;> cases hq : q x <;>
        simp [hp, hq] at * <;> omega

abbrev Index := Nat
def indices : List Index := List.range 48
def triples : List (Index × Index × Index) := cart indices (cart indices indices)
def collision (t : Index × Index × Index) : Bool :=
  (t.1 == t.2.1) || (t.1 == t.2.2) || (t.2.1 == t.2.2)
def distinctTriples : List (Index × Index × Index) := triples.filter (fun t => !collision t)

theorem length_indices : indices.length = 48 := by simp [indices]
theorem length_triples : triples.length = 110592 := by
  simp [triples, length_cart, length_indices]

theorem count_index_eq_le (i : Index) : indices.countP (fun j => i == j) ≤ 1 := by
  have h := List.count_range (a := i) (n := 48)
  have heq : (fun j => i == j) = (fun j => j == i) := by
    funext j
    exact Bool.beq_comm
  rw [heq]
  change (List.range 48).count i ≤ 1
  rw [h]
  split <;> omega

theorem count_collision_le : triples.countP collision ≤ 6912 := by
  have h12 : triples.countP (fun t => t.1 == t.2.1) ≤ 2304 := by
    apply count_cart_le indices (cart indices indices) _ 48
    intro i _
    change (cart indices indices).countP (fun y => i == y.1) ≤ 48
    rw [count_cart_left indices indices (fun j => i == j), length_indices]
    have := count_index_eq_le i
    omega
  have h13 : triples.countP (fun t => t.1 == t.2.2) ≤ 2304 := by
    apply count_cart_le indices (cart indices indices) _ 48
    intro i _
    change (cart indices indices).countP (fun y => i == y.2) ≤ 48
    rw [count_cart_right indices indices (fun j => i == j), length_indices]
    have := count_index_eq_le i
    omega
  have h23 : triples.countP (fun t => t.2.1 == t.2.2) ≤ 2304 := by
    change (cart indices (cart indices indices)).countP (fun t => t.2.1 == t.2.2) ≤ 2304
    rw [count_cart_right indices (cart indices indices) (fun t => t.1 == t.2), length_indices]
    have h : (cart indices indices).countP (fun t => t.1 == t.2) ≤ 48 := by
      apply count_cart_le indices indices _ 1
      intro i _
      exact count_index_eq_le i
    omega
  have hA := count_union_le triples (fun t => t.1 == t.2.1) (fun t => t.1 == t.2.2)
  have hB := count_union_le triples
    (fun t => (t.1 == t.2.1) || (t.1 == t.2.2)) (fun t => t.2.1 == t.2.2)
  unfold collision
  omega

theorem majority_exists (g : Index → Bool) :
    ∃ b : Bool, 24 ≤ indices.countP (fun i => g i == b) := by
  have h := List.length_eq_countP_add_countP g (l := indices)
  have hn : (fun i => decide (¬g i = true)) = (fun i => !g i) := by
    funext i
    cases g i <;> rfl
  rw [hn] at h
  rw [length_indices] at h
  by_cases ht : 24 ≤ indices.countP g
  · exact ⟨true, by simpa using ht⟩
  · refine ⟨false, ?_⟩
    have : 24 ≤ indices.countP (fun i => !g i) := by omega
    simpa using this

def allAgree (p q r : Index → Bool) (t : Index × Index × Index) : Bool :=
  p t.1 && (q t.2.1 && r t.2.2)

theorem count_agree_distinct (p q r : Index → Bool)
    (hp : 24 ≤ indices.countP p) (hq : 24 ≤ indices.countP q)
    (hr : 24 ≤ indices.countP r) :
    6912 ≤ distinctTriples.countP (allAgree p q r) := by
  have hprod := count_cart indices (cart indices indices) p
    (fun t => q t.1 && r t.2)
  rw [count_cart] at hprod
  have h1 : 24 * 24 ≤ indices.countP q * indices.countP r := Nat.mul_le_mul hq hr
  have h2 : 24 * (24 * 24) ≤ indices.countP p *
      (indices.countP q * indices.countP r) := Nat.mul_le_mul hp h1
  have hremove := count_remove triples (allAgree p q r) collision
  have hcollision := count_collision_le
  change triples.countP (allAgree p q r) = _ at hprod
  unfold distinctTriples
  rw [List.countP_filter]
  omega

structure Equation (Name : Type) where
  first : Name
  second : Name
  third : Name
  rhs : Bool

variable {Name : Type}

def satisfied (e : Equation Name) (g : Name → Bool) : Bool :=
  (xor (xor (g e.first) (g e.second)) (g e.third)) == e.rhs

def clone (e : Equation Name) (t : Index × Index × Index) : Equation (Name × Index) :=
  ⟨(e.first, t.1), (e.second, t.2.1), (e.third, t.2.2), e.rhs⟩

theorem lift_preserves (e : Equation Name) (t : Index × Index × Index) (g : Name → Bool) :
    satisfied (clone e t) (fun z => g z.1) = satisfied e g := rfl

theorem clone_names_distinct (e : Equation Name) (t : Index × Index × Index)
    (h : collision t = false) :
    (clone e t).first ≠ (clone e t).second ∧
    (clone e t).first ≠ (clone e t).third ∧
    (clone e t).second ≠ (clone e t).third := by
  simp only [collision, Bool.or_eq_false_iff, beq_eq_false_iff_ne] at h
  dsimp [clone]
  exact ⟨fun he => h.1.1 (congrArg Prod.snd he),
    fun he => h.1.2 (congrArg Prod.snd he),
    fun he => h.2 (congrArg Prod.snd he)⟩

theorem agree_failure (e : Equation Name) (g : Name × Index → Bool)
    (majority : Name → Bool) (t : Index × Index × Index)
    (hfail : satisfied e majority = false)
    (h : allAgree (fun i => g (e.first, i) == majority e.first)
      (fun i => g (e.second, i) == majority e.second)
      (fun i => g (e.third, i) == majority e.third) t = true) :
    satisfied (clone e t) g = false := by
  simp only [allAgree, Bool.and_eq_true, beq_iff_eq] at h
  simpa [satisfied, clone, h.1, h.2.1, h.2.2] using hfail

theorem failed_occurrence_clones (e : Equation Name) (g : Name × Index → Bool)
    (majority : Name → Bool)
    (hmaj : ∀ v, 24 ≤ indices.countP (fun i => g (v, i) == majority v))
    (hfail : satisfied e majority = false) :
    6912 ≤ distinctTriples.countP (fun t => !satisfied (clone e t) g) := by
  have h := count_agree_distinct
    (fun i => g (e.first, i) == majority e.first)
    (fun i => g (e.second, i) == majority e.second)
    (fun i => g (e.third, i) == majority e.third)
    (hmaj _) (hmaj _) (hmaj _)
  apply Nat.le_trans h
  apply List.countP_mono_left
  intro t _ ht
  have hf := agree_failure e g majority t hfail ht
  simp [hf]

def majorityBit (g : Index → Bool) : Bool := decide (24 ≤ indices.countP g)

theorem majorityBit_count (g : Index → Bool) :
    24 ≤ indices.countP (fun i => g i == majorityBit g) := by
  have h := List.length_eq_countP_add_countP g (l := indices)
  have hn : (fun i => decide (¬g i = true)) = (fun i => !g i) := by
    funext i
    cases g i <;> rfl
  rw [hn, length_indices] at h
  by_cases ht : 24 ≤ indices.countP g
  · simp [majorityBit, ht]
  · have hn : 24 ≤ indices.countP (fun i => !g i) := by omega
    simpa [majorityBit, ht] using hn

def cloneList (source : List (Equation Name)) : List (Equation (Name × Index)) :=
  source.flatMap fun e => distinctTriples.map (clone e)

theorem length_cloneList (source : List (Equation Name)) :
    (cloneList source).length = source.length * distinctTriples.length := by
  induction source with
  | nil => simp [cloneList]
  | cons e es ih =>
      simp only [cloneList, List.flatMap_cons, List.length_append, List.length_map] at *
      rw [ih]
      simp [Nat.add_mul, Nat.add_comm]

theorem cloneList_failure_lower (source : List (Equation Name))
    (g : Name × Index → Bool) (majority : Name → Bool)
    (hmaj : ∀ v, 24 ≤ indices.countP (fun i => g (v, i) == majority v)) :
    6912 * source.countP (fun e => !satisfied e majority) ≤
      (cloneList source).countP (fun e => !satisfied e g) := by
  induction source with
  | nil => simp [cloneList]
  | cons e es ih =>
      simp only [cloneList, List.flatMap_cons, List.countP_append,
        List.countP_map, Function.comp_def] at *
      cases hf : satisfied e majority with
      | false =>
          have he := failed_occurrence_clones e g majority hmaj hf
          simp [hf, Nat.mul_add] at *
          omega
      | true =>
          simp [hf]
          omega

theorem clone_gap (source : List (Equation Name))
    (source_gap : ∀ A : Name → Bool,
      source.length ≤ 4 * source.countP (fun e => !satisfied e A))
    (g : Name × Index → Bool) :
    (cloneList source).length ≤
      64 * (cloneList source).countP (fun e => !satisfied e g) := by
  let majority : Name → Bool := fun v => majorityBit (fun i => g (v, i))
  have hmaj : ∀ v, 24 ≤ indices.countP (fun i => g (v, i) == majority v) :=
    fun v => majorityBit_count (fun i => g (v, i))
  have hlocal := cloneList_failure_lower source g majority hmaj
  have hsource := source_gap majority
  have hd : distinctTriples.length ≤ 110592 := by
    exact Nat.le_trans (List.length_filter_le _ triples) (by simp [length_triples])
  rw [length_cloneList]
  have hlength := Nat.mul_le_mul_left source.length hd
  have hs := Nat.mul_le_mul_right 27648 hsource
  have hc := Nat.mul_le_mul_left 64 hlocal
  omega

theorem clone_completeness_count (source : List (Equation Name)) (A : Name → Bool) :
    (cloneList source).countP (fun e => satisfied e (fun z => A z.1)) =
      source.countP (fun e => satisfied e A) * distinctTriples.length := by
  induction source with
  | nil => simp [cloneList]
  | cons e es ih =>
      simp only [cloneList, List.flatMap_cons, List.countP_append,
        List.countP_map, Function.comp_def] at *
      rw [ih]
      have he : (fun t => satisfied (clone e t) (fun z => A z.1)) =
          (fun _ => satisfied e A) := by
        funext t
        exact lift_preserves e t A
      rw [he, count_const]
      cases h : satisfied e A <;> simp [h, Nat.add_mul, Nat.add_comm]

end DFVSGames.Reduction.CloneGap


namespace DFVSGames.Foundations.Hastad.FoldedEquation

open DFVSGames.Reduction.CloneGap
open scoped BigOperators

variable {I J : Type} [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]

abbrev Address (i₀ : I) (j₀ : J) := HalfCube i₀ ⊕ HalfCube j₀

def storedAssignment {i₀ : I} {j₀ : J}
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool) :
    Address i₀ j₀ → Bool := Sum.elim tableA tableB

def rhsCorrection (π : J → I) (i₀ : I) (j₀ : J)
    (f : Cube I) (μ : Cube J) : Bool := f i₀ ^^ f (π j₀) ^^ μ j₀

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem correction_eq (π : J → I) (i₀ : I) (j₀ : J)
    (f : Cube I) (g μ : Cube J) :
    (f i₀ ^^ g j₀ ^^ thirdQuery π f g μ j₀) = rhsCorrection π i₀ j₀ f μ := by
  change (f i₀ ^^ g j₀ ^^ (g j₀ ^^ (f (π j₀) ^^ μ j₀))) =
    (f i₀ ^^ f (π j₀) ^^ μ j₀)
  cases f i₀ <;> cases g j₀ <;> cases f (π j₀) <;> cases μ j₀ <;> rfl

def equation (π : J → I) (i₀ : I) (j₀ : J)
    (f : Cube I) (g μ : Cube J) : Equation (Address i₀ j₀) where
  first := .inl (canonicalInput i₀ f)
  second := .inr (canonicalInput j₀ g)
  third := .inr (canonicalInput j₀ (thirdQuery π f g μ))
  rhs := rhsCorrection π i₀ j₀ f μ

theorem xor_corrections (a b c u v w : Bool) :
    ((a ^^ b ^^ c) == (u ^^ v ^^ w)) =
      !((a ^^ u) ^^ (b ^^ v) ^^ (c ^^ w)) := by
  cases a <;> cases b <;> cases c <;> cases u <;> cases v <;> cases w <;> rfl

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in

theorem equation_satisfied (π : J → I) (i₀ : I) (j₀ : J)
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J) :
    satisfied (equation π i₀ j₀ f g μ) (storedAssignment tableA tableB) =
      !(foldedAnswer i₀ tableA f ^^ foldedAnswer j₀ tableB g ^^
        foldedAnswer j₀ tableB (thirdQuery π f g μ)) := by
  change ((tableA (canonicalInput i₀ f) ^^ tableB (canonicalInput j₀ g) ^^
      tableB (canonicalInput j₀ (thirdQuery π f g μ))) == rhsCorrection π i₀ j₀ f μ) = _
  rw [← correction_eq π i₀ j₀ f g μ]
  exact xor_corrections _ _ _ _ _ _

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem equation_satisfied_iff (π : J → I) (i₀ : I) (j₀ : J)
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J) :
    satisfied (equation π i₀ j₀ f g μ) (storedAssignment tableA tableB) = true ↔
      (foldedAnswer i₀ tableA f ^^ foldedAnswer j₀ tableB g ^^
        foldedAnswer j₀ tableB (thirdQuery π f g μ)) = false := by
  rw [equation_satisfied]
  simp

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem equation_indicator (π : J → I) (i₀ : I) (j₀ : J)
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J) :
    (if satisfied (equation π i₀ j₀ f g μ) (storedAssignment tableA tableB)
      then (1 : ℝ) else 0) =
    (if foldedAnswer i₀ tableA f ^^ foldedAnswer j₀ tableB g ^^
      foldedAnswer j₀ tableB (thirdQuery π f g μ) then 0 else 1) := by
  rw [equation_satisfied]
  cases foldedAnswer i₀ tableA f ^^ foldedAnswer j₀ tableB g ^^
    foldedAnswer j₀ tableB (thirdQuery π f g μ) <;> rfl

theorem satisfied_repeated_right {Name : Type} (e : Equation Name)
    (assignment : Name → Bool) (h : e.second = e.third) :
    satisfied e assignment = (assignment e.first == e.rhs) := by
  unfold satisfied
  rw [← h]
  cases assignment e.first <;> cases assignment e.second <;> cases e.rhs <;> rfl

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem equation_repeated_right (π : J → I) (i₀ : I) (j₀ : J)
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J)
    (h : canonicalInput j₀ g = canonicalInput j₀ (thirdQuery π f g μ)) :
    satisfied (equation π i₀ j₀ f g μ) (storedAssignment tableA tableB) =
      (tableA (canonicalInput i₀ f) == rhsCorrection π i₀ j₀ f μ) := by
  apply satisfied_repeated_right
  exact congrArg Sum.inr h

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem restrict_thirdQuery (valid : J → Bool) (π : J → I)
    (f : Cube I) (g μ : Cube J) :
    restrictQuery valid (thirdQuery π f g μ) =
      thirdQuery (fun j : {j : J // valid j = true} => π j.val) f
        (restrictQuery valid g) (restrictQuery valid μ) := rfl

def conditionedEquation (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true})
    (f : Cube I) (g μ : Cube J) : Equation (Address i₀ j₀) :=
  equation (fun j => π j.val) i₀ j₀ f (restrictQuery valid g) (restrictQuery valid μ)

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem conditionedEquation_rhs (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true}) (f : Cube I) (g μ : Cube J) :
    (conditionedEquation valid π i₀ j₀ f g μ).rhs =
      (f i₀ ^^ f (π j₀.val) ^^ μ j₀.val) := rfl

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem conditionedEquation_satisfied (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true})
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J) :
    satisfied (conditionedEquation valid π i₀ j₀ f g μ) (storedAssignment tableA tableB) =
      !(foldedAnswer i₀ tableA f ^^ conditionedFoldedAnswer valid j₀ tableB g ^^
        conditionedFoldedAnswer valid j₀ tableB (thirdQuery π f g μ)) := by
  unfold conditionedEquation
  rw [equation_satisfied]
  simp only [conditionedFoldedAnswer, restrict_thirdQuery]

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem conditionedEquation_satisfied_iff (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true})
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J) :
    satisfied (conditionedEquation valid π i₀ j₀ f g μ) (storedAssignment tableA tableB) = true ↔
      (foldedAnswer i₀ tableA f ^^ conditionedFoldedAnswer valid j₀ tableB g ^^
        conditionedFoldedAnswer valid j₀ tableB (thirdQuery π f g μ)) = false := by
  rw [conditionedEquation_satisfied]
  simp

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem conditionedEquation_indicator (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true})
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J) :
    (if satisfied (conditionedEquation valid π i₀ j₀ f g μ) (storedAssignment tableA tableB)
      then (1 : ℝ) else 0) =
    (if foldedAnswer i₀ tableA f ^^ conditionedFoldedAnswer valid j₀ tableB g ^^
      conditionedFoldedAnswer valid j₀ tableB (thirdQuery π f g μ) then 0 else 1) := by
  rw [conditionedEquation_satisfied]
  cases foldedAnswer i₀ tableA f ^^ conditionedFoldedAnswer valid j₀ tableB g ^^
    conditionedFoldedAnswer valid j₀ tableB (thirdQuery π f g μ) <;> rfl

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in
theorem conditionedEquation_repeated_right (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true})
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool)
    (f : Cube I) (g μ : Cube J)
    (h : canonicalInput j₀ (restrictQuery valid g) =
      canonicalInput j₀ (restrictQuery valid (thirdQuery π f g μ))) :
    satisfied (conditionedEquation valid π i₀ j₀ f g μ) (storedAssignment tableA tableB) =
      (tableA (canonicalInput i₀ f) == (f i₀ ^^ f (π j₀.val) ^^ μ j₀.val)) := by
  apply satisfied_repeated_right
  exact congrArg Sum.inr h

noncomputable section

def equationAcceptance (ε : ℝ) (π : J → I) (i₀ : I) (j₀ : J)
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool) : ℝ :=
  𝔼 f, ∑ μ, noiseWeight ε μ *
    (𝔼 g, if satisfied (equation π i₀ j₀ f g μ) (storedAssignment tableA tableB)
      then (1 : ℝ) else 0)

theorem equationAcceptance_eq (ε : ℝ) (π : J → I) (i₀ : I) (j₀ : J)
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool) :
    equationAcceptance ε π i₀ j₀ tableA tableB =
      testAcceptance ε π (foldedAnswer i₀ tableA) (foldedAnswer j₀ tableB) := by
  unfold equationAcceptance testAcceptance
  simp_rw [equation_indicator]

def conditionedEquationAcceptance (ε : ℝ) (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true})
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool) : ℝ :=
  𝔼 f, ∑ μ, noiseWeight ε μ *
    (𝔼 g, if satisfied (conditionedEquation valid π i₀ j₀ f g μ)
      (storedAssignment tableA tableB) then (1 : ℝ) else 0)

theorem conditionedEquationAcceptance_eq (ε : ℝ) (valid : J → Bool) (π : J → I)
    (i₀ : I) (j₀ : {j : J // valid j = true})
    (tableA : HalfCube i₀ → Bool) (tableB : HalfCube j₀ → Bool) :
    conditionedEquationAcceptance ε valid π i₀ j₀ tableA tableB =
      testAcceptance ε π (foldedAnswer i₀ tableA)
        (conditionedFoldedAnswer valid j₀ tableB) := by
  unfold conditionedEquationAcceptance testAcceptance
  simp_rw [conditionedEquation_indicator]

end

end DFVSGames.Foundations.Hastad.FoldedEquation


namespace DFVSGames.Foundations.Hastad

open DFVSGames.Reduction.CloneGap
open scoped BigOperators

def mapEquation {Name Name' : Type} (rename : Name → Name')
    (e : Equation Name) : Equation Name' where
  first := rename e.first
  second := rename e.second
  third := rename e.third
  rhs := e.rhs

theorem satisfied_mapEquation {Name Name' : Type} (rename : Name → Name')
    (e : Equation Name) (assignment : Name' → Bool) :
    satisfied (mapEquation rename e) assignment = satisfied e (assignment ∘ rename) := rfl

namespace EmptyContext

variable {I : Type} [Fintype I] [DecidableEq I]

abbrev Address (i₀ : I) := HalfCube i₀ ⊕ Unit

def storedAssignment {i₀ : I} (table : HalfCube i₀ → Bool) (dummy : Bool) :
    Address i₀ → Bool := Sum.elim table (fun _ => dummy)

def equation (i₀ : I) (f : Cube I) : Equation (Address i₀) where
  first := .inl (canonicalInput i₀ f)
  second := .inr ()
  third := .inr ()
  rhs := f i₀

omit [Fintype I] [DecidableEq I] in
@[simp] theorem equation_second_eq_third (i₀ : I) (f : Cube I) :
    (equation i₀ f).second = (equation i₀ f).third := rfl

omit [Fintype I] [DecidableEq I] in

theorem equation_satisfied (i₀ : I) (assignment : Address i₀ → Bool) (f : Cube I) :
    satisfied (equation i₀ f) assignment =
      !(foldedAnswer i₀ (fun h => assignment (.inl h)) f) := by
  rw [FoldedEquation.satisfied_repeated_right (equation i₀ f) assignment rfl]
  change (assignment (.inl (canonicalInput i₀ f)) == f i₀) =
    !(assignment (.inl (canonicalInput i₀ f)) ^^ f i₀)
  cases assignment (.inl (canonicalInput i₀ f)) <;> cases f i₀ <;> rfl

omit [Fintype I] [DecidableEq I] in
theorem equation_storedAssignment_satisfied (i₀ : I) (table : HalfCube i₀ → Bool)
    (dummy : Bool) (f : Cube I) :
    satisfied (equation i₀ f) (storedAssignment table dummy) =
      !(foldedAnswer i₀ table f) := equation_satisfied i₀ _ f

omit [Fintype I] [DecidableEq I] in
theorem equation_flip_first (i₀ : I) (f : Cube I) :
    (equation i₀ (cubeFlip f)).first = (equation i₀ f).first := by
  change Sum.inl (canonicalInput i₀ (cubeFlip f)) = Sum.inl (canonicalInput i₀ f)
  rw [canonicalInput_flip]

omit [Fintype I] [DecidableEq I] in
theorem equation_flip_rhs (i₀ : I) (f : Cube I) :
    (equation i₀ (cubeFlip f)).rhs = !(equation i₀ f).rhs := rfl

omit [Fintype I] [DecidableEq I] in
theorem equation_satisfied_flip (i₀ : I) (assignment : Address i₀ → Bool)
    (f : Cube I) :
    satisfied (equation i₀ (cubeFlip f)) assignment =
      !(satisfied (equation i₀ f) assignment) := by
  rw [equation_satisfied, foldedAnswer_flip, equation_satisfied]

def pairedEquations (i₀ : I) (f : Cube I) : List (Equation (Address i₀)) :=
  [equation i₀ f, equation i₀ (cubeFlip f)]

omit [Fintype I] [DecidableEq I] in
@[simp] theorem pairedEquations_length (i₀ : I) (f : Cube I) :
    (pairedEquations i₀ f).length = 2 := rfl

omit [Fintype I] [DecidableEq I] in
theorem pairedEquations_ne_nil (i₀ : I) (f : Cube I) :
    pairedEquations i₀ f ≠ [] := by simp [pairedEquations]

omit [Fintype I] [DecidableEq I] in
theorem pairedEquations_acceptedCount (i₀ : I) (assignment : Address i₀ → Bool)
    (f : Cube I) :
    (pairedEquations i₀ f).countP (fun e => satisfied e assignment) = 1 := by
  simp only [pairedEquations, List.countP_cons, List.countP_nil, equation_satisfied_flip]
  cases satisfied (equation i₀ f) assignment <;> rfl

omit [Fintype I] [DecidableEq I] in
theorem pairedEquations_failureCount (i₀ : I) (assignment : Address i₀ → Bool)
    (f : Cube I) :
    (pairedEquations i₀ f).countP (fun e => !(satisfied e assignment)) = 1 := by
  simp only [pairedEquations, List.countP_cons, List.countP_nil, equation_satisfied_flip]
  cases satisfied (equation i₀ f) assignment <;> rfl

noncomputable section

def equationAcceptance (i₀ : I) (assignment : Address i₀ → Bool) : ℝ :=
  𝔼 f : Cube I, if satisfied (equation i₀ f) assignment then (1 : ℝ) else 0

def equationBias (i₀ : I) (assignment : Address i₀ → Bool) : ℝ :=
  𝔼 f : Cube I, if satisfied (equation i₀ f) assignment then (1 : ℝ) else -1

theorem foldedMean_eq_zero (i₀ : I) (table : HalfCube i₀ → Bool) :
    (𝔼 f : Cube I, bitSign (foldedAnswer i₀ table f)) = 0 := by
  simpa [coefficient, walsh, bitSign] using foldedAnswer_zero_coefficient i₀ table

omit [Fintype I] [DecidableEq I] in
theorem equation_indicator (i₀ : I) (assignment : Address i₀ → Bool) (f : Cube I) :
    (if satisfied (equation i₀ f) assignment then (1 : ℝ) else 0) =
      (1 + bitSign (foldedAnswer i₀ (fun h => assignment (.inl h)) f)) / 2 := by
  rw [equation_satisfied]
  cases foldedAnswer i₀ (fun h => assignment (.inl h)) f <;> norm_num [bitSign]

theorem equationAcceptance_eq_half (i₀ : I) (assignment : Address i₀ → Bool) :
    equationAcceptance i₀ assignment = 1 / 2 := by
  unfold equationAcceptance
  simp_rw [equation_indicator, div_eq_mul_inv, ← Finset.expect_mul,
    Finset.expect_add_distrib, Fintype.expect_const, foldedMean_eq_zero]
  norm_num

theorem equationBias_eq_zero (i₀ : I) (assignment : Address i₀ → Bool) :
    equationBias i₀ assignment = 0 := by
  unfold equationBias
  have hi (f : Cube I) :
      (if satisfied (equation i₀ f) assignment then (1 : ℝ) else -1) =
        bitSign (foldedAnswer i₀ (fun h => assignment (.inl h)) f) := by
    rw [equation_satisfied]
    cases foldedAnswer i₀ (fun h => assignment (.inl h)) f <;> norm_num [bitSign]
  simp_rw [hi]
  exact foldedMean_eq_zero i₀ _

theorem mapped_equationAcceptance_eq_half {Name : Type} (i₀ : I)
    (rename : Address i₀ → Name) (assignment : Name → Bool) :
    (𝔼 f : Cube I, if satisfied (mapEquation rename (equation i₀ f)) assignment
      then (1 : ℝ) else 0) = 1 / 2 := by
  simp only [satisfied_mapEquation]
  exact equationAcceptance_eq_half i₀ (assignment ∘ rename)

end
end EmptyContext
end DFVSGames.Foundations.Hastad


namespace DFVSGames.Reduction.FiniteNoise

open scoped BigOperators
open DFVSGames.Foundations.Hastad (noiseWeight)

def realizedBit {D : Nat} (z : Fin D) : Bool := decide (z.val = 0)

def realizedNoise {A : Type*} {D : Nat} (z : A → Fin D) : A → Bool :=
  fun a => realizedBit (z a)

theorem realizedBit_eq_true_iff {D : Nat} (positive : 0 < D) (z : Fin D) :
    realizedBit z = true ↔ z = ⟨0, positive⟩ := by
  simp [realizedBit, Fin.ext_iff]

theorem coordinate_fiber_sum {D : Nat} (positive : 0 < D) (b : Bool) :
    (∑ z : Fin D, if realizedBit z = b then (D : ℝ)⁻¹ else 0) =
      if b then (D : ℝ)⁻¹ else 1 - (D : ℝ)⁻¹ := by
  have hd : (D : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt positive
  have ht : (∑ z : Fin D, if realizedBit z = true then (D : ℝ)⁻¹ else 0) =
      (D : ℝ)⁻¹ := by
    simp_rw [realizedBit_eq_true_iff positive]
    simp
  have htotal :
      (∑ z : Fin D, if realizedBit z = false then (D : ℝ)⁻¹ else 0) +
      (∑ z : Fin D, if realizedBit z = true then (D : ℝ)⁻¹ else 0) = 1 := by
    rw [← Finset.sum_add_distrib]
    have he (z : Fin D) :
        (if realizedBit z = false then (D : ℝ)⁻¹ else 0) +
          (if realizedBit z = true then (D : ℝ)⁻¹ else 0) = (D : ℝ)⁻¹ := by
      cases realizedBit z <;> simp
    simp_rw [he]
    simp [hd]
  cases b <;> simp only [Bool.false_eq_true, ite_false, ite_true]
  · linarith
  · exact ht

variable {A : Type*} [Fintype A] [DecidableEq A]

theorem noiseWeight_eq_fiber_sum {D : Nat} (positive : 0 < D) (μ : A → Bool) :
    noiseWeight ((D : ℝ)⁻¹) μ =
      ∑ z : A → Fin D,
        ∏ a, if realizedBit (z a) = μ a then (D : ℝ)⁻¹ else 0 := by
  rw [← Fintype.prod_sum (fun (a : A) (z : Fin D) =>
    if realizedBit z = μ a then (D : ℝ)⁻¹ else 0)]
  unfold noiseWeight
  apply Finset.prod_congr rfl
  intro a _
  exact (coordinate_fiber_sum positive (μ a)).symm

omit [DecidableEq A] in
theorem fiber_product_eq_indicator {D : Nat} (z : A → Fin D) (μ : A → Bool) :
    (∏ a, if realizedBit (z a) = μ a then (D : ℝ)⁻¹ else 0) =
      if realizedNoise z = μ then ((D : ℝ)⁻¹) ^ Fintype.card A else 0 := by
  classical
  by_cases h : realizedNoise z = μ
  · rw [ite_eq_left h]
    have ha (a : A) : realizedBit (z a) = μ a := congrFun h a
    simp [ha]
  · rw [ite_eq_right h]
    obtain ⟨a, ha⟩ := Function.ne_iff.mp h
    apply Finset.prod_eq_zero (Finset.mem_univ a)
    exact ite_eq_right ha

theorem expect_realizedNoise {D : Nat} (positive : 0 < D) (H : (A → Bool) → ℝ) :
    (𝔼 z : A → Fin D, H (realizedNoise z)) =
      ∑ μ : A → Bool, noiseWeight ((D : ℝ)⁻¹) μ * H μ := by
  classical
  symm
  simp_rw [noiseWeight_eq_fiber_sum positive, Finset.sum_mul]
  rw [Finset.sum_comm]
  simp_rw [fiber_product_eq_indicator]
  have he (z : A → Fin D) :
      (∑ μ : A → Bool,
        (if realizedNoise z = μ then ((D : ℝ)⁻¹) ^ Fintype.card A else 0) * H μ) =
      ((D : ℝ)⁻¹) ^ Fintype.card A * H (realizedNoise z) := by
    simp [eq_comm]
  simp_rw [he]
  rw [Fintype.expect_eq_sum_div_card]
  simp only [Fintype.card_fun, Fintype.card_fin, Nat.cast_pow]
  rw [div_eq_mul_inv, inv_pow]
  rw [← Finset.mul_sum]
  exact mul_comm _ _

end DFVSGames.Reduction.FiniteNoise


noncomputable section

namespace DFVSGames.Foundations.Hastad

open scoped BigOperators
open DFVSGames.Reduction.FiniteNoise

variable {I J : Type*} [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]

omit [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] in

theorem dictator_test_parity (π : J → I) (i : I) (j : J) (hπ : π j = i)
    (f : Cube I) (g μ : Cube J) :
    (f i ^^ g j ^^ (thirdQuery π f g μ) j) = μ j := by
  simp only [thirdQuery, cubeXor, hπ]
  cases f i <;> cases g j <;> cases μ j <;> rfl

theorem noise_coordinate_sign (ε : ℝ) (j : J) :
    (∑ μ : Cube J, noiseWeight ε μ * bitSign (μ j)) = 1 - 2 * ε := by
  have hs : support (coordinateMask j) = {j} := by
    ext k
    simp [support, coordinateMask]
  have h := noise_walsh ε (coordinateMask j)
  simp_rw [walsh_symm (coordinateMask j), walsh_coordinateMask] at h
  simpa only [hs, Finset.card_singleton, pow_one] using h

theorem noise_coordinate_false (ε : ℝ) (j : J) :
    (∑ μ : Cube J, noiseWeight ε μ * if μ j then (0 : ℝ) else 1) = 1 - ε := by
  have hind (b : Bool) : (if b then (0 : ℝ) else 1) =
      (1 + bitSign b) * (2 : ℝ)⁻¹ := by
    cases b <;> norm_num [bitSign]
  have ht (μ : Cube J) : noiseWeight ε μ * (if μ j then (0 : ℝ) else 1) =
      (noiseWeight ε μ + noiseWeight ε μ * bitSign (μ j)) * (2 : ℝ)⁻¹ := by
    rw [hind]
    ring
  simp_rw [ht]
  rw [← Finset.sum_mul, Finset.sum_add_distrib, noiseWeight_sum, noise_coordinate_sign]
  ring

theorem testAcceptance_dictator (ε : ℝ) (π : J → I)
    (i : I) (j : J) (hπ : π j = i) :
    testAcceptance ε π (fun f => f i) (fun g => g j) = 1 - ε := by
  have hg (f : Cube I) (μ : Cube J) :
      (𝔼 g : Cube J, if f i ^^ g j ^^ (thirdQuery π f g μ) j then (0 : ℝ) else 1) =
        if μ j then 0 else 1 := by
    calc
      _ = 𝔼 _g : Cube J, if μ j then (0 : ℝ) else 1 := by
        apply Finset.expect_congr rfl
        intro g _
        rw [dictator_test_parity π i j hπ f g μ]
      _ = _ := Fintype.expect_const _
  change (𝔼 f : Cube I, ∑ μ : Cube J, noiseWeight ε μ *
    (𝔼 g : Cube J, if f i ^^ g j ^^ (thirdQuery π f g μ) j then (0 : ℝ) else 1)) = _
  simp_rw [hg]
  rw [Fintype.expect_const]
  exact noise_coordinate_false ε j

theorem testAcceptance_folded_dictators (ε : ℝ) (π : J → I)
    (i₀ i : I) (j₀ j : J) (hπ : π j = i) :
    testAcceptance ε π (foldedAnswer i₀ (fun h => h.val i))
      (foldedAnswer j₀ (fun h => h.val j)) = 1 - ε := by
  have hA : foldedAnswer i₀ (fun h => h.val i) = (fun f : Cube I => f i) :=
    funext (foldedAnswer_dictator i₀ i)
  have hB : foldedAnswer j₀ (fun h => h.val j) = (fun g : Cube J => g j) :=
    funext (foldedAnswer_dictator j₀ j)
  rw [hA, hB]
  exact testAcceptance_dictator ε π i j hπ

theorem testAcceptance_conditioned_dictators (ε : ℝ) (π : J → I)
    (validI : I → Bool) (validJ : J → Bool)
    (i₀ i : {i : I // validI i = true}) (j₀ j : {j : J // validJ j = true})
    (hπ : π j.val = i.val) :
    testAcceptance ε π
      (conditionedFoldedAnswer validI i₀ (fun h => h.val i))
      (conditionedFoldedAnswer validJ j₀ (fun h => h.val j)) = 1 - ε := by
  have hA : conditionedFoldedAnswer validI i₀ (fun h => h.val i) =
      (fun f : Cube I => f i.val) :=
    funext (conditionedFoldedAnswer_dictator validI i₀ i)
  have hB : conditionedFoldedAnswer validJ j₀ (fun h => h.val j) =
      (fun g : Cube J => g j.val) :=
    funext (conditionedFoldedAnswer_dictator validJ j₀ j)
  rw [hA, hB]
  exact testAcceptance_dictator ε π i.val j.val hπ

theorem testAcceptance_folded_conditioned_dictators (ε : ℝ) (π : J → I)
    (i₀ i : I) (valid : J → Bool)
    (j₀ j : {j : J // valid j = true}) (hπ : π j.val = i) :
    testAcceptance ε π (foldedAnswer i₀ (fun h => h.val i))
      (conditionedFoldedAnswer valid j₀ (fun h => h.val j)) = 1 - ε := by
  have hA : foldedAnswer i₀ (fun h => h.val i) = (fun f : Cube I => f i) :=
    funext (foldedAnswer_dictator i₀ i)
  have hB : conditionedFoldedAnswer valid j₀ (fun h => h.val j) =
      (fun g : Cube J => g j.val) :=
    funext (conditionedFoldedAnswer_dictator valid j₀ j)
  rw [hA, hB]
  exact testAcceptance_dictator ε π i j.val hπ

def realizedTestAcceptance (D : Nat) (π : J → I)
    (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  𝔼 f : Cube I, 𝔼 z : J → Fin D, 𝔼 g : Cube J,
    if A f ^^ B g ^^ B (thirdQuery π f g (realizedNoise z)) then 0 else 1

theorem realizedTestAcceptance_eq_testAcceptance {D : Nat} (positive : 0 < D)
    (π : J → I) (A : Cube I → Bool) (B : Cube J → Bool) :
    realizedTestAcceptance D π A B = testAcceptance ((D : ℝ)⁻¹) π A B := by
  unfold realizedTestAcceptance testAcceptance
  apply Finset.expect_congr rfl
  intro f _
  exact expect_realizedNoise positive
    (fun μ => 𝔼 g : Cube J, if A f ^^ B g ^^ B (thirdQuery π f g μ) then 0 else 1)

theorem realizedTestAcceptance_dictator {D : Nat} (positive : 0 < D)
    (π : J → I) (i : I) (j : J) (hπ : π j = i) :
    realizedTestAcceptance D π (fun f => f i) (fun g => g j) = 1 - (D : ℝ)⁻¹ := by
  rw [realizedTestAcceptance_eq_testAcceptance positive]
  exact testAcceptance_dictator _ π i j hπ

theorem realizedTestAcceptance_folded_dictators {D : Nat} (positive : 0 < D)
    (π : J → I) (i₀ i : I) (j₀ j : J) (hπ : π j = i) :
    realizedTestAcceptance D π (foldedAnswer i₀ (fun h => h.val i))
      (foldedAnswer j₀ (fun h => h.val j)) = 1 - (D : ℝ)⁻¹ := by
  rw [realizedTestAcceptance_eq_testAcceptance positive]
  exact testAcceptance_folded_dictators _ π i₀ i j₀ j hπ

theorem realizedTestAcceptance_conditioned_dictators {D : Nat} (positive : 0 < D)
    (π : J → I) (validI : I → Bool) (validJ : J → Bool)
    (i₀ i : {i : I // validI i = true}) (j₀ j : {j : J // validJ j = true})
    (hπ : π j.val = i.val) :
    realizedTestAcceptance D π
      (conditionedFoldedAnswer validI i₀ (fun h => h.val i))
      (conditionedFoldedAnswer validJ j₀ (fun h => h.val j)) = 1 - (D : ℝ)⁻¹ := by
  rw [realizedTestAcceptance_eq_testAcceptance positive]
  exact testAcceptance_conditioned_dictators _ π validI validJ i₀ i j₀ j hπ

theorem realizedTestAcceptance_folded_conditioned_dictators {D : Nat} (positive : 0 < D)
    (π : J → I) (i₀ i : I) (valid : J → Bool)
    (j₀ j : {j : J // valid j = true}) (hπ : π j.val = i) :
    realizedTestAcceptance D π (foldedAnswer i₀ (fun h => h.val i))
      (conditionedFoldedAnswer valid j₀ (fun h => h.val j)) = 1 - (D : ℝ)⁻¹ := by
  rw [realizedTestAcceptance_eq_testAcceptance positive]
  exact testAcceptance_folded_conditioned_dictators _ π i₀ i valid j₀ j hπ

end DFVSGames.Foundations.Hastad
end

end OAI