Datasets:

roseking
/

openAI-math-dataset

import OAI.Computability.BinPacking.Information.GameDecoder
import OAI.Computability.BinPacking.Reductions.CloneGap
namespace OAI

noncomputable section

namespace BinPackingGames.Foundations.Hastad.FoldedEquation
open BinPackingGames.Reduction.CloneGap
open scoped BigOperators
variable {I J : Type}

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
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

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 _ _ _ _ _ _

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

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

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

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)

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

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]

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

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

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

variable [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]
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 BinPackingGames.Foundations.Hastad.FoldedEquation
namespace BinPackingGames.Foundations.Hastad
open BinPackingGames.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}

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

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

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

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

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]

theorem equation_flip_rhs (i₀ : I) (f : Cube I) :
    (equation i₀ (cubeFlip f)).rhs = !(equation i₀ f).rhs := rfl

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)]

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

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

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

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

variable [Fintype I] [DecidableEq I]

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 EmptyContext
end BinPackingGames.Foundations.Hastad
end

end OAI

	import OAI.Computability.BinPacking.Information.GameDecoder
	import OAI.Computability.BinPacking.Reductions.CloneGap

	namespace OAI

	noncomputable section

	namespace BinPackingGames.Foundations.Hastad.FoldedEquation

	open BinPackingGames.Reduction.CloneGap
	open scoped BigOperators

	variable {I J : Type}

	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

	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

	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 _ _ _ _ _ _

	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

	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

	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

	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)

	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

	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]

	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

	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

	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

	variable [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]

	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 BinPackingGames.Foundations.Hastad.FoldedEquation

	namespace BinPackingGames.Foundations.Hastad

	open BinPackingGames.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}

	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

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

	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

	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

	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]

	theorem equation_flip_rhs (i₀ : I) (f : Cube I) :
	(equation i₀ (cubeFlip f)).rhs = !(equation i₀ f).rhs := rfl

	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)]

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

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

	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

	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

	variable [Fintype I] [DecidableEq I]

	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 EmptyContext
	end BinPackingGames.Foundations.Hastad

	end

	end OAI