Datasets:

roseking
/

openAI-math-dataset

import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Ring.GeomSum
import Mathlib.Algebra.Ring.Parity
import Mathlib.Basic.Real.Basic
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import OAI.Computability.BinPacking.Games.Reindexing
import OAI.Computability.BinPacking.PCP.BinaryFormula
namespace OAI

noncomputable section

namespace BinPackingGames.Foundations.Hastad
section

open scoped BigOperators
open Finset

abbrev Cube (I : Type*) := I → Bool
def bitSign (b : Bool) : ℝ := if b then -1 else 1

@[simp] theorem bitSign_sq (b : Bool) : bitSign b ^ 2 = 1 := by
  cases b <;> norm_num [bitSign]

theorem bitSign_xor (b c : Bool) :
    bitSign (b ^^ c) = bitSign b * bitSign c := by
  cases b <;> cases c <;> norm_num [bitSign]

def cubeXor {I : Type*} (x y : Cube I) : Cube I := fun i => x i ^^ y i

def walsh {I : Type*} [Fintype I] [DecidableEq I] (s x : Cube I) : ℝ :=
  ∏ i, bitSign (s i && x i)

theorem walsh_symm {I : Type*} [Fintype I] [DecidableEq I] (s x : Cube I) :
    walsh s x = walsh x s := by
  simp [walsh, Bool.and_comm]
@[simp] theorem walsh_sq {I : Type*} [Fintype I] [DecidableEq I] (s x : Cube I) :
    walsh s x ^ 2 = 1 := by
  simp [walsh, ← Finset.prod_pow]

theorem walsh_xor {I : Type*} [Fintype I] [DecidableEq I] (s x y : Cube I) :
    walsh s (cubeXor x y) = walsh s x * walsh s y := by
  simp only [walsh, ← Finset.prod_mul_distrib]
  apply Finset.prod_congr rfl
  intro i _
  simp only [cubeXor]
  cases s i <;> cases x i <;> cases y i <;> norm_num [bitSign]

theorem coordinate_orthogonality (s t : Bool) :
    (∑ x : Bool, bitSign (s && x) * bitSign (t && x)) =
      if s = t then 2 else 0 := by
  cases s <;> cases t <;> norm_num [bitSign, Fintype.sum_bool]

theorem walsh_sum_orthogonality {I : Type*} [Fintype I] [DecidableEq I]
    (s t : Cube I) :
    (∑ x, walsh s x * walsh t x) =
      if s = t then (Fintype.card (Cube I) : ℝ) else 0 := by
  classical
  simp only [walsh, ← Finset.prod_mul_distrib]
  rw [← Fintype.prod_sum (fun (i : I) (b : Bool) =>
    bitSign (s i && b) * bitSign (t i && b))]
  simp only [coordinate_orthogonality]
  by_cases h : s = t
  · subst t
    simp
  · rw [ite_eq_right h]
    obtain ⟨i, hi⟩ := Function.ne_iff.mp h
    apply Finset.prod_eq_zero (Finset.mem_univ i)
    exact ite_eq_right hi

theorem walsh_orthogonality {I : Type*} [Fintype I] [DecidableEq I]
    (s t : Cube I) :
    (𝔼 x, walsh s x * walsh t x) = if s = t then 1 else 0 := by
  classical
  rw [Fintype.expect_eq_sum_div_card, walsh_sum_orthogonality]
  split_ifs <;> simp

def coefficient {I : Type*} [Fintype I] [DecidableEq I] (f : Cube I → ℝ) (s : Cube I) : ℝ :=
  𝔼 x, f x * walsh s x

theorem walsh_inversion {I : Type*} [Fintype I] [DecidableEq I] (f : Cube I → ℝ) (x : Cube I) :
    (∑ s, coefficient f s * walsh s x) = f x := by
  classical
  unfold coefficient
  simp_rw [Finset.expect_mul]
  rw [← Finset.expect_sum_comm]
  have h (y : Cube I) :
      (∑ s, (f y * walsh s y) * walsh s x) =
        f y * (if y = x then (Fintype.card (Cube I) : ℝ) else 0) := by
    simp only [mul_assoc, ← Finset.mul_sum]
    congr 1
    simp_rw [walsh_symm (x := y), walsh_symm (x := x)]
    exact walsh_sum_orthogonality y x
  simp_rw [h]
  rw [Fintype.expect_eq_sum_div_card]
  simp

theorem walsh_inner {I : Type*} [Fintype I] [DecidableEq I] (f g : Cube I → ℝ) :
    (∑ s, coefficient f s * coefficient g s) = 𝔼 x, f x * g x := by
  classical
  unfold coefficient
  simp_rw [Finset.mul_expect]
  rw [← Finset.expect_sum_comm]
  apply Finset.expect_congr rfl
  intro x _
  have h : (∑ s, (𝔼 y, f y * walsh s y) * (g x * walsh s x)) =
      g x * (∑ s, coefficient f s * walsh s x) := by
    simp only [Finset.mul_sum, coefficient]
    apply Finset.sum_congr rfl
    intro s _
    ring
  rw [h, walsh_inversion]
  ring

theorem walsh_parseval {I : Type*} [Fintype I] [DecidableEq I] (f : Cube I → ℝ) :
    (∑ s, coefficient f s ^ 2) = 𝔼 x, f x ^ 2 := by
  simpa [pow_two] using walsh_inner f f

theorem sign_parseval {I : Type*} [Fintype I] [DecidableEq I] (f : Cube I → Bool) :
    (∑ s, coefficient (fun x => bitSign (f x)) s ^ 2) = 1 := by
  rw [walsh_parseval]
  simp

open scoped BigOperators
open Finset

variable {I J : Type*} [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]
def support (s : Cube I) : Finset I := Finset.univ.filter fun i => s i = true

def noiseWeight (ε : ℝ) (μ : Cube I) : ℝ :=
  ∏ i, if μ i then ε else 1 - ε

omit [DecidableEq I] in
theorem noiseWeight_nonneg {ε : ℝ} (hε : 0 ≤ ε) (hε' : ε ≤ 1) (μ : Cube I) :
    0 ≤ noiseWeight ε μ := by
  apply Finset.prod_nonneg
  intro i _
  split <;> linarith
theorem noiseWeight_sum (ε : ℝ) : (∑ μ : Cube I, noiseWeight ε μ) = 1 := by
  unfold noiseWeight
  rw [← Fintype.prod_sum (fun (_ : I) (b : Bool) => if b then ε else 1 - ε)]
  simp
theorem noise_walsh (ε : ℝ) (s : Cube I) :
    (∑ μ, noiseWeight ε μ * walsh s μ) = (1 - 2 * ε) ^ (support s).card := by
  unfold noiseWeight walsh
  simp only [← Finset.prod_mul_distrib]
  rw [← Fintype.prod_sum (fun i b =>
    (if b then ε else 1 - ε) * bitSign (s i && b))]
  have hc (i : I) : (∑ b : Bool,
      (if b then ε else 1 - ε) * bitSign (s i && b)) =
        if s i then 1 - 2 * ε else 1 := by
    cases s i <;> simp [bitSign]; ring
  simp_rw [hc]
  simp [support, Finset.prod_ite]
def thirdQuery (π : J → I) (f : Cube I) (g μ : Cube J) : Cube J :=
  cubeXor g (cubeXor (fun y => f (π y)) μ)
theorem walsh_autocorrelation (B : Cube J → ℝ) (h : Cube J) :
    (𝔼 g, B g * B (cubeXor g h)) =
      ∑ s, coefficient B s ^ 2 * walsh s h := by
  conv_lhs =>
    enter [2, g, 2]
    rw [← walsh_inversion B (cubeXor g h)]
  simp_rw [Finset.mul_sum, walsh_xor]
  rw [Finset.expect_sum_comm]
  apply Finset.sum_congr rfl
  intro s _
  have hterm : (fun g => B g * (coefficient B s * (walsh s g * walsh s h))) =
      (fun g => (coefficient B s * walsh s h) * (B g * walsh s g)) := by
    funext g
    ring
  rw [hterm, ← Finset.mul_expect]
  change (coefficient B s * walsh s h) * coefficient B s = _
  ring
def projectedCoefficient (π : J → I) (A : Cube I → ℝ) (s : Cube J) : ℝ :=
  𝔼 f, A f * walsh s (fun y => f (π y))
def testBias (ε : ℝ) (π : J → I) (A : Cube I → ℝ) (B : Cube J → ℝ) : ℝ :=
  𝔼 f, ∑ μ, noiseWeight ε μ * A f *
    (𝔼 g, B g * B (thirdQuery π f g μ))
theorem testBias_fourier (ε : ℝ) (π : J → I)
    (A : Cube I → ℝ) (B : Cube J → ℝ) :
    testBias ε π A B = ∑ s,
      projectedCoefficient π A s * coefficient B s ^ 2 *
        (1 - 2 * ε) ^ (support s).card := by
  unfold testBias thirdQuery
  simp_rw [walsh_autocorrelation, Finset.mul_sum]
  conv_lhs =>
    enter [2, f]
    rw [Finset.sum_comm]
  rw [Finset.expect_sum_comm]
  apply Finset.sum_congr rfl
  intro s _
  have hterm (f : Cube I) :
      (∑ μ, noiseWeight ε μ * A f *
        (coefficient B s ^ 2 * walsh s (cubeXor (fun y => f (π y)) μ))) =
      (A f * walsh s (fun y => f (π y))) * coefficient B s ^ 2 *
        (1 - 2 * ε) ^ (support s).card := by
    simp_rw [walsh_xor]
    have heq (μ : Cube J) : noiseWeight ε μ * A f *
        (coefficient B s ^ 2 * (walsh s (fun y => f (π y)) * walsh s μ)) =
      ((A f * walsh s (fun y => f (π y))) * coefficient B s ^ 2) *
        (noiseWeight ε μ * walsh s μ) := by ring
    simp_rw [heq]
    rw [← Finset.mul_sum, noise_walsh]
  simp_rw [hterm, ← Finset.expect_mul]
  rfl
def testAcceptance (ε : ℝ) (π : J → I)
    (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  𝔼 f, ∑ μ, noiseWeight ε μ *
    (𝔼 g, if A f ^^ B g ^^ B (thirdQuery π f g μ) then 0 else 1)
theorem threeBit_indicator (a b c : Bool) :
    (if a ^^ b ^^ c then (0 : ℝ) else 1) =
      (1 + bitSign a * bitSign b * bitSign c) / 2 := by
  cases a <;> cases b <;> cases c <;> norm_num [bitSign]
theorem testAcceptance_eq (ε : ℝ) (π : J → I)
    (A : Cube I → Bool) (B : Cube J → Bool) :
    testAcceptance ε π A B =
      (1 + testBias ε π (fun f => bitSign (A f)) (fun g => bitSign (B g))) / 2 := by
  unfold testAcceptance testBias
  simp_rw [threeBit_indicator, div_eq_mul_inv,
    ← Finset.expect_mul, Finset.expect_add_distrib]
  simp only [Fintype.expect_const]
  have hterm (f : Cube I) (μ : Cube J) :
      noiseWeight ε μ * ((1 + 𝔼 g,
        bitSign (A f) * bitSign (B g) * bitSign (B (thirdQuery π f g μ))) * 2⁻¹) =
      (noiseWeight ε μ + noiseWeight ε μ * bitSign (A f) *
        (𝔼 g, bitSign (B g) * bitSign (B (thirdQuery π f g μ)))) * 2⁻¹ := by
    simp_rw [mul_assoc (bitSign (A f)), ← Finset.mul_expect]
    ring
  simp_rw [hterm, ← Finset.sum_mul, Finset.sum_add_distrib, noiseWeight_sum,
    ← Finset.expect_mul, Finset.expect_add_distrib]
  simp
end
section
open scoped BigOperators
open Finset
variable {I : Type*} [Fintype I] [DecidableEq I]
def cubeFlip (f : Cube I) : Cube I := fun i => !(f i)
omit [Fintype I] [DecidableEq I] in
@[simp] theorem cubeFlip_cubeFlip (f : Cube I) : cubeFlip (cubeFlip f) = f := by
  funext i
  simp [cubeFlip]
@[simp] theorem bitSign_not (b : Bool) : bitSign (!b) = -bitSign b := by
  cases b <;> norm_num [bitSign]
theorem sum_cubeFlip (F : Cube I → ℝ) : (∑ f, F (cubeFlip f)) = ∑ f, F f := by
  refine Finset.sum_bij (fun f _ => cubeFlip f) ?_ ?_ ?_ ?_
  · intro f _
    exact Finset.mem_univ _
  · intro f _ g _ h
    have h' := congrArg cubeFlip h
    simpa only [cubeFlip_cubeFlip] using h'
  · intro g _
    exact ⟨cubeFlip g, Finset.mem_univ _, cubeFlip_cubeFlip g⟩
  · intro f _
    rfl
theorem coefficient_zero_of_odd (F : Cube I → ℝ)
    (hF : ∀ f, F (cubeFlip f) = -F f) :
    coefficient F (fun _ => false) = 0 := by
  have hs : (∑ f, F f) = 0 := by
    have h := sum_cubeFlip F
    simp_rw [hF, Finset.sum_neg_distrib] at h
    linarith
  simp [coefficient, walsh, bitSign, Fintype.expect_eq_sum_div_card, hs]
theorem coefficient_sign_zero_of_folded (A : Cube I → Bool)
    (hA : ∀ f, A (cubeFlip f) = !(A f)) :
    coefficient (fun f => bitSign (A f)) (fun _ => false) = 0 := by
  apply coefficient_zero_of_odd
  intro f
  rw [hA, bitSign_not]
def representative (i₀ : I) (f : Cube I) : Cube I :=
  if f i₀ then cubeFlip f else f
omit [Fintype I] [DecidableEq I] in
theorem representative_at (i₀ : I) (f : Cube I) :
    representative i₀ f i₀ = false := by
  cases hf : f i₀ <;> simp [representative, cubeFlip, hf]
omit [Fintype I] [DecidableEq I] in
theorem representative_flip (i₀ : I) (f : Cube I) :
    representative i₀ (cubeFlip f) = representative i₀ f := by
  cases hf : f i₀ <;> simp [representative, cubeFlip, hf]
abbrev HalfCube (i₀ : I) := {f : Cube I // f i₀ = false}
def canonicalInput (i₀ : I) (f : Cube I) : HalfCube i₀ :=
  ⟨representative i₀ f, representative_at i₀ f⟩
omit [Fintype I] [DecidableEq I] in
theorem canonicalInput_flip (i₀ : I) (f : Cube I) :
    canonicalInput i₀ (cubeFlip f) = canonicalInput i₀ f := by
  apply Subtype.ext
  exact representative_flip i₀ f
def foldedAnswer (i₀ : I) (table : HalfCube i₀ → Bool) (f : Cube I) : Bool :=
  table (canonicalInput i₀ f) ^^ f i₀
omit [Fintype I] [DecidableEq I] in
theorem foldedAnswer_flip (i₀ : I) (table : HalfCube i₀ → Bool) (f : Cube I) :
    foldedAnswer i₀ table (cubeFlip f) = !(foldedAnswer i₀ table f) := by
  unfold foldedAnswer
  rw [canonicalInput_flip]
  change (table (canonicalInput i₀ f) ^^ !(f i₀)) =
    !(table (canonicalInput i₀ f) ^^ f i₀)
  cases table (canonicalInput i₀ f) <;> cases f i₀ <;> rfl
theorem foldedAnswer_zero_coefficient (i₀ : I) (table : HalfCube i₀ → Bool) :
    coefficient (fun f => bitSign (foldedAnswer i₀ table f)) (fun _ => false) = 0 :=
  coefficient_sign_zero_of_folded _ (foldedAnswer_flip i₀ table)
omit [Fintype I] [DecidableEq I] in
theorem foldedAnswer_dictator (i₀ i : I) (f : Cube I) :
    foldedAnswer i₀ (fun h => h.val i) f = f i := by
  cases hf : f i₀ <;> cases hi : f i <;>
    simp [foldedAnswer, canonicalInput, representative, cubeFlip, hf, hi]
def coordinateMask (i : I) : Cube I := fun j => decide (j = i)
def cubeToggle (i : I) (f : Cube I) : Cube I :=
  cubeXor f (coordinateMask i)
omit [Fintype I] in
@[simp] theorem cubeToggle_twice (i : I) (f : Cube I) :
    cubeToggle i (cubeToggle i f) = f := by
  funext j
  change ((f j ^^ coordinateMask i j) ^^ coordinateMask i j) = f j
  cases f j <;> cases coordinateMask i j <;> rfl
theorem walsh_coordinateMask (s : Cube I) (i : I) :
    walsh s (coordinateMask i) = bitSign (s i) := by
  unfold walsh
  rw [Finset.prod_eq_single i]
  · simp [coordinateMask]
  · intro j _ hj
    simp [coordinateMask, hj, bitSign]
  · intro h
    exact False.elim (h (Finset.mem_univ _))
theorem walsh_cubeToggle (s : Cube I) (i : I) (f : Cube I) (hs : s i = true) :
    walsh s (cubeToggle i f) = -walsh s f := by
  rw [cubeToggle, walsh_xor, walsh_coordinateMask, hs]
  simp [bitSign]
theorem sum_cubeToggle (i : I) (F : Cube I → ℝ) :
    (∑ f, F (cubeToggle i f)) = ∑ f, F f := by
  refine Finset.sum_bij (fun f _ => cubeToggle i f) ?_ ?_ ?_ ?_
  · intro f _
    exact Finset.mem_univ _
  · intro f _ g _ h
    have h' := congrArg (cubeToggle i) h
    simpa only [cubeToggle_twice] using h'
  · intro g _
    exact ⟨cubeToggle i g, Finset.mem_univ _, cubeToggle_twice i g⟩
  · intro f _
    rfl
def restrictQuery (valid : I → Bool) (f : Cube I) :
    Cube {i : I // valid i = true} := fun i => f i.val
omit [Fintype I] in
theorem restrictQuery_cubeToggle (valid : I → Bool) (i : I)
    (hi : valid i = false) (f : Cube I) :
    restrictQuery valid (cubeToggle i f) = restrictQuery valid f := by
  funext j
  have hj : j.val ≠ i := by
    intro heq
    have hp := j.property
    rw [heq, hi] at hp
    contradiction
  simp [restrictQuery, cubeToggle, cubeXor, coordinateMask, hj]
theorem conditioned_coefficient_zero_invalid (valid : I → Bool)
    (B : Cube {i : I // valid i = true} → ℝ) (s : Cube I) (i : I)
    (hi : valid i = false) (hs : s i = true) :
    coefficient (fun f => B (restrictQuery valid f)) s = 0 := by
  have hsum := sum_cubeToggle i (fun f => B (restrictQuery valid f) * walsh s f)
  simp_rw [restrictQuery_cubeToggle valid i hi, walsh_cubeToggle s i _ hs,
    mul_neg, Finset.sum_neg_distrib] at hsum
  have hz : (∑ f, B (restrictQuery valid f) * walsh s f) = 0 := by linarith
  simp [coefficient, Fintype.expect_eq_sum_div_card, hz]
theorem conditioned_coefficient_support (valid : I → Bool)
    (B : Cube {i : I // valid i = true} → ℝ) (s : Cube I)
    (hne : coefficient (fun f => B (restrictQuery valid f)) s ≠ 0) :
    ∀ i ∈ support s, valid i = true := by
  intro i hi
  have hs := (Finset.mem_filter.mp hi).2
  cases hv : valid i
  · exact False.elim (hne (conditioned_coefficient_zero_invalid valid B s i hv hs))
  · rfl
def conditionedFoldedAnswer (valid : I → Bool) (i₀ : {i : I // valid i = true})
    (table : HalfCube i₀ → Bool) (f : Cube I) : Bool :=
  foldedAnswer i₀ table (restrictQuery valid f)
omit [Fintype I] [DecidableEq I] in
theorem conditionedFoldedAnswer_flip (valid : I → Bool)
    (i₀ : {i : I // valid i = true}) (table : HalfCube i₀ → Bool) (f : Cube I) :
    conditionedFoldedAnswer valid i₀ table (cubeFlip f) =
      !(conditionedFoldedAnswer valid i₀ table f) := by
  change foldedAnswer i₀ table (cubeFlip (restrictQuery valid f)) = _
  exact foldedAnswer_flip i₀ table (restrictQuery valid f)
theorem conditionedFoldedAnswer_zero_coefficient (valid : I → Bool)
    (i₀ : {i : I // valid i = true}) (table : HalfCube i₀ → Bool) :
    coefficient (fun f => bitSign (conditionedFoldedAnswer valid i₀ table f))
      (fun _ => false) = 0 :=
  coefficient_sign_zero_of_folded _ (conditionedFoldedAnswer_flip valid i₀ table)
omit [Fintype I] [DecidableEq I] in
theorem conditionedFoldedAnswer_dictator (valid : I → Bool)
    (i₀ i : {i : I // valid i = true}) (f : Cube I) :
    conditionedFoldedAnswer valid i₀ (fun h => h.val i) f = f i.val := by
  exact foldedAnswer_dictator i₀ i (restrictQuery valid f)
end
open scoped BigOperators
open Finset
variable {I J : Type*} [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]
def parityProjection (π : J → I) (s : Cube J) : Cube I := fun i =>
  decide (Odd ((support s).filter fun y => π y = i).card)
theorem walsh_support (s x : Cube I) :
    walsh s x = ∏ i ∈ support s, bitSign (x i) := by
  unfold walsh support
  rw [Finset.prod_filter]
  apply Finset.prod_congr rfl
  intro i _
  cases s i <;> simp [bitSign]
theorem sign_power_parity (b : Bool) (n : Nat) :
    bitSign b ^ n = bitSign (decide (Odd n) && b) := by
  cases b
  · simp [bitSign]
  · simp only [Bool.and_true, bitSign, ite_true, decide_eq_true_eq]
    rw [neg_one_pow_eq_ite]
    by_cases h : Even n
    · simp [h, Nat.not_odd_iff_even.mpr h]
    · simp [h, Nat.not_even_iff_odd.mp h]
theorem walsh_pullback (π : J → I) (s : Cube J) (f : Cube I) :
    walsh s (fun y => f (π y)) = walsh (parityProjection π s) f := by
  rw [walsh_support]
  rw [← Finset.prod_fiberwise' (support s) π (fun i => bitSign (f i))]
  unfold walsh parityProjection
  apply Finset.prod_congr rfl
  intro i _
  simp only [Finset.prod_const]
  exact sign_power_parity (f i) _

theorem projectedCoefficient_eq (π : J → I) (A : Cube I → ℝ) (s : Cube J) :
    projectedCoefficient π A s = coefficient A (parityProjection π s) := by
  simp only [projectedCoefficient, coefficient, walsh_pullback]

omit [DecidableEq J] in
theorem projection_has_preimage (π : J → I) (s : Cube J) (i : I)
    (hi : i ∈ support (parityProjection π s)) :
    ∃ y ∈ support s, π y = i := by
  have hodd : Odd ((support s).filter fun y => π y = i).card := by
    have hbit := (Finset.mem_filter.mp hi).2
    simpa only [parityProjection, decide_eq_true_eq] using hbit
  have hp : 0 < ((support s).filter fun y => π y = i).card := by
    obtain ⟨n, hn⟩ := hodd
    omega
  obtain ⟨y, hy⟩ := Finset.card_pos.mp hp
  exact ⟨y, (Finset.mem_filter.mp hy).1, (Finset.mem_filter.mp hy).2⟩

theorem scaled_geometric_power_le_one {t : ℝ} (ht : 0 ≤ t) (ht' : t ≤ 1) (n : Nat) :
    (n : ℝ) * (1 - t) * t ^ n ≤ 1 := by
  have hsum : (n : ℝ) * t ^ n ≤ ∑ j ∈ Finset.range n, t ^ j := by
    calc
      (n : ℝ) * t ^ n = ∑ _j ∈ Finset.range n, t ^ n := by simp
      _ ≤ ∑ j ∈ Finset.range n, t ^ j := by
        apply Finset.sum_le_sum
        intro j hj
        exact pow_le_pow_of_le_one ht ht' (Nat.le_of_lt (Finset.mem_range.mp hj))
  have hm := mul_le_mul_of_nonneg_right hsum (sub_nonneg.mpr ht')
  rw [geom_sum_mul_neg] at hm
  have hp : 0 ≤ t ^ n := pow_nonneg ht _
  nlinarith
theorem noise_decay_bound {ε : ℝ} (hε : 0 < ε) (hε' : ε ≤ 1 / 2)
    {k : Nat} (hk : 0 < k) :
    4 * ε * ((1 - 2 * ε) ^ k) ^ 2 ≤ 1 / (k : ℝ) := by
  have h := scaled_geometric_power_le_one (t := 1 - 2 * ε)
    (by linarith) (by linarith) (2 * k)
  have hkp : (0 : ℝ) < k := Nat.cast_pos.mpr hk
  apply (le_div_iff₀ hkp).mpr
  have hpow : (1 - 2 * ε) ^ (2 * k) = ((1 - 2 * ε) ^ k) ^ 2 := by
    rw [Nat.mul_comm, pow_mul]
  rw [hpow] at h
  norm_num only [Nat.cast_mul, Nat.cast_ofNat] at h
  nlinarith

def zeroMask : Cube I := fun _ => false

omit [DecidableEq I] in
theorem support_eq_empty_iff (s : Cube I) : support s = ∅ ↔ s = zeroMask := by
  constructor
  · intro h
    funext i
    have hi : i ∉ support s := by rw [h]; simp
    have : s i ≠ true := by simpa [support] using hi
    cases hs : s i <;> simp_all [zeroMask]
  · intro h
    subst s
    simp [zeroMask, support]

omit [Fintype I] [DecidableEq J] in
theorem parityProjection_zero (π : J → I) :
    parityProjection π (zeroMask : Cube J) = zeroMask := by
  funext i
  simp [parityProjection, support, zeroMask]

def agreementProbability (π : J → I) (a : Cube I) (b : Cube J) : ℝ :=
  (∑ i ∈ support a, (((support b).filter fun y => π y = i).card : ℝ)) /
    ((support a).card * (support b).card : ℝ)

omit [DecidableEq J] in
theorem agreementProbability_nonneg (π : J → I) (a : Cube I) (b : Cube J) :
    0 ≤ agreementProbability π a b := by
  unfold agreementProbability
  apply div_nonneg
  · exact Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _)
  · exact mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)

omit [DecidableEq J] in
theorem agreementProbability_projection (π : J → I) (b : Cube J)
    (ha : (support (parityProjection π b)).Nonempty) :
    1 / ((support b).card : ℝ) ≤ agreementProbability π (parityProjection π b) b := by
  have hca : (0 : ℝ) < (support (parityProjection π b)).card :=
    Nat.cast_pos.mpr (Finset.card_pos.mpr ha)
  have hb : (support b).Nonempty := by
    obtain ⟨i, hi⟩ := ha
    obtain ⟨y, hy, _⟩ := projection_has_preimage π b i hi
    exact ⟨y, hy⟩
  have hcb : (0 : ℝ) < (support b).card := Nat.cast_pos.mpr (Finset.card_pos.mpr hb)
  have hsum : ((support (parityProjection π b)).card : ℝ) ≤
      ∑ i ∈ support (parityProjection π b),
        (((support b).filter fun y => π y = i).card : ℝ) := by
    calc
      _ = ∑ _i ∈ support (parityProjection π b), (1 : ℝ) := by simp
      _ ≤ _ := by
        apply Finset.sum_le_sum
        intro i hi
        obtain ⟨y, hy, hπ⟩ := projection_has_preimage π b i hi
        have hp : 0 < ((support b).filter fun y => π y = i).card :=
          Finset.card_pos.mpr ⟨y, Finset.mem_filter.mpr ⟨hy, hπ⟩⟩
        exact_mod_cast hp
  unfold agreementProbability
  apply (div_le_div_iff₀ hcb (mul_pos hca hcb)).mpr
  nlinarith

def decoderSuccess (π : J → I) (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  ∑ b, coefficient (fun g => bitSign (B g)) b ^ 2 *
    (∑ a, coefficient (fun f => bitSign (A f)) a ^ 2 * agreementProbability π a b)
def decodingEnergy (π : J → I) (A : Cube I → ℝ) (B : Cube J → ℝ) : ℝ :=
  ∑ b, projectedCoefficient π A b ^ 2 * coefficient B b ^ 2 / (support b).card
theorem decodingEnergy_le_success (π : J → I) (A : Cube I → Bool) (B : Cube J → Bool)
    (hzero : coefficient (fun f => bitSign (A f)) zeroMask = 0) :
    decodingEnergy π (fun f => bitSign (A f)) (fun g => bitSign (B g)) ≤
      decoderSuccess π A B := by
  unfold decodingEnergy decoderSuccess
  apply Finset.sum_le_sum
  intro b _
  rw [projectedCoefficient_eq]
  have hsum :
      coefficient (fun f => bitSign (A f)) (parityProjection π b) ^ 2 *
        agreementProbability π (parityProjection π b) b ≤
      ∑ a, coefficient (fun f => bitSign (A f)) a ^ 2 * agreementProbability π a b := by
    apply Finset.single_le_sum (s := Finset.univ)
      (f := fun a => coefficient (fun f => bitSign (A f)) a ^ 2 * agreementProbability π a b)
    · intro a _
      exact mul_nonneg (sq_nonneg _) (agreementProbability_nonneg π a b)
    · exact Finset.mem_univ _
  have hm := mul_le_mul_of_nonneg_left hsum
    (sq_nonneg (coefficient (fun g => bitSign (B g)) b))
  by_cases ha : (support (parityProjection π b)).Nonempty
  · have hagree := agreementProbability_projection π b ha
    have ht := mul_le_mul_of_nonneg_left hagree
      (mul_nonneg (sq_nonneg (coefficient (fun f => bitSign (A f))
        (parityProjection π b))) (sq_nonneg (coefficient (fun g => bitSign (B g)) b)))
    have heq : coefficient (fun f => bitSign (A f)) (parityProjection π b) ^ 2 *
        coefficient (fun g => bitSign (B g)) b ^ 2 / (support b).card =
      (coefficient (fun f => bitSign (A f)) (parityProjection π b) ^ 2 *
        coefficient (fun g => bitSign (B g)) b ^ 2) * (1 / (support b).card) := by ring
    rw [heq]
    exact ht.trans (by convert hm using 1; ring)
  · have hempty : support (parityProjection π b) = ∅ := Finset.not_nonempty_iff_eq_empty.mp ha
    have hz := (support_eq_empty_iff _).mp hempty
    simpa [hz, hzero] using hm

theorem bias_sq_le_weighted_energy (ε : ℝ) (π : J → I)
    (A : Cube I → ℝ) (B : Cube J → Bool) :
    testBias ε π A (fun g => bitSign (B g)) ^ 2 ≤
      ∑ b, (projectedCoefficient π A b * coefficient (fun g => bitSign (B g)) b *
        (1 - 2 * ε) ^ (support b).card) ^ 2 := by
  rw [testBias_fourier]
  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
    (fun b => projectedCoefficient π A b * coefficient (fun g => bitSign (B g)) b *
      (1 - 2 * ε) ^ (support b).card)
    (fun b => coefficient (fun g => bitSign (B g)) b)
  rw [sign_parseval, mul_one] at h
  convert h using 1
  congr 1
  apply Finset.sum_congr rfl
  intro b _
  ring

theorem decoder_four_epsilon_bias_sq (ε : ℝ) (π : J → I)
    (A : Cube I → Bool) (B : Cube J → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2)
    (hzero : coefficient (fun f => bitSign (A f)) zeroMask = 0) :
    4 * ε * testBias ε π (fun f => bitSign (A f)) (fun g => bitSign (B g)) ^ 2 ≤
      decoderSuccess π A B := by
  apply le_trans _ (decodingEnergy_le_success π A B hzero)
  have hcs := mul_le_mul_of_nonneg_left
    (bias_sq_le_weighted_energy ε π (fun f => bitSign (A f)) B)
    (by linarith : 0 ≤ 4 * ε)
  apply hcs.trans
  rw [Finset.mul_sum]
  unfold decodingEnergy
  apply Finset.sum_le_sum
  intro b _
  by_cases hk : 0 < (support b).card
  · have hd := noise_decay_bound hε hε' hk
    have hm := mul_le_mul_of_nonneg_left hd
      (mul_nonneg (sq_nonneg (projectedCoefficient π (fun f => bitSign (A f)) b))
        (sq_nonneg (coefficient (fun g => bitSign (B g)) b)))
    convert hm using 1 <;> ring
  · have hb : support b = ∅ := Finset.card_eq_zero.mp (by omega)
    have hbzero := (support_eq_empty_iff b).mp hb
    subst b
    simp [projectedCoefficient_eq, parityProjection_zero, hzero, support, zeroMask]

theorem folded_decoder_bound (ε : ℝ) (π : J → I)
    (i₀ : I) (table : HalfCube i₀ → Bool) (B : Cube J → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * testBias ε π (fun f => bitSign (foldedAnswer i₀ table f))
      (fun g => bitSign (B g)) ^ 2 ≤ decoderSuccess π (foldedAnswer i₀ table) B := by
  exact decoder_four_epsilon_bias_sq ε π (foldedAnswer i₀ table) B hε hε'
    (foldedAnswer_zero_coefficient i₀ table)

def validAgreementProbability (valid : J → Bool) (π : J → I)
    (a : Cube I) (b : Cube J) : ℝ :=
  (∑ i ∈ support a,
    (((support b).filter fun y => π y = i ∧ valid y = true).card : ℝ)) /
    ((support a).card * (support b).card : ℝ)

omit [DecidableEq J] in
theorem validAgreementProbability_eq (valid : J → Bool) (π : J → I)
    (a : Cube I) (b : Cube J) (hb : ∀ y ∈ support b, valid y = true) :
    validAgreementProbability valid π a b = agreementProbability π a b := by
  unfold validAgreementProbability agreementProbability
  congr 1
  apply Finset.sum_congr rfl
  intro i _
  congr 1
  apply congrArg Finset.card
  apply Finset.filter_congr
  intro y hy
  simp [hb y hy]

def validDecoderSuccess (valid : J → Bool) (π : J → I)
    (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  ∑ b, coefficient (fun g => bitSign (B g)) b ^ 2 *
    (∑ a, coefficient (fun f => bitSign (A f)) a ^ 2 *
      validAgreementProbability valid π a b)
theorem conditioned_validDecoderSuccess (valid : J → Bool) (π : J → I)
    (A : Cube I → Bool) (j₀ : {j : J // valid j = true})
    (table : HalfCube j₀ → Bool) :
    validDecoderSuccess valid π A (conditionedFoldedAnswer valid j₀ table) =
      decoderSuccess π A (conditionedFoldedAnswer valid j₀ table) := by
  unfold validDecoderSuccess decoderSuccess
  apply Finset.sum_congr rfl
  intro b _
  by_cases hc : coefficient (fun g => bitSign (conditionedFoldedAnswer valid j₀ table g)) b = 0
  · simp [hc]
  · have hb : ∀ y ∈ support b, valid y = true := by
      apply conditioned_coefficient_support valid (fun g => bitSign (foldedAnswer j₀ table g)) b
      exact hc
    congr 1
    apply Finset.sum_congr rfl
    intro a _
    rw [validAgreementProbability_eq valid π a b hb]

theorem conditioned_folded_decoder_bound (ε : ℝ) (π : J → I) (valid : J → Bool)
    (i₀ : I) (tableA : HalfCube i₀ → Bool)
    (j₀ : {j : J // valid j = true}) (tableB : HalfCube j₀ → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * testBias ε π (fun f => bitSign (foldedAnswer i₀ tableA f))
      (fun g => bitSign (conditionedFoldedAnswer valid j₀ tableB g)) ^ 2 ≤
      validDecoderSuccess valid π (foldedAnswer i₀ tableA)
        (conditionedFoldedAnswer valid j₀ tableB) := by
  rw [conditioned_validDecoderSuccess]
  exact folded_decoder_bound ε π i₀ tableA
    (conditionedFoldedAnswer valid j₀ tableB) hε hε'

end BinPackingGames.Foundations.Hastad
namespace BinPackingGames.Foundations.Hastad.SourceContexts
open Target PCP

def clauseAnswerEquiv : ClauseAnswer ≃ Bool × Bool × Bool where
  toFun a := (a.first, a.second, a.third)
  invFun a := ⟨a.1, a.2.1, a.2.2⟩
  left_inv a := by cases a; rfl
  right_inv a := by rcases a with ⟨a, b, c⟩; rfl

def clauseAnswerFinEquiv : ClauseAnswer ≃ Fin 8 :=
  clauseAnswerEquiv.trans
    ((Equiv.prodCongr finTwoEquiv.symm
      (Equiv.prodCongr finTwoEquiv.symm finTwoEquiv.symm)).trans
      ((Equiv.prodCongr (Equiv.refl (Fin 2)) finProdFinEquiv).trans finProdFinEquiv))

instance clauseAnswerFintype : Fintype ClauseAnswer :=
  Fintype.ofEquiv (Bool × Bool × Bool) clauseAnswerEquiv.symm

instance clauseAnswerInhabited : Inhabited ClauseAnswer := ⟨⟨false, false, false⟩⟩

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

instance slotInhabited : Inhabited Slot := ⟨.first⟩

@[simp] theorem card_clauseAnswer : Fintype.card ClauseAnswer = 8 := by
  rw [Fintype.card_congr clauseAnswerFinEquiv]
  rfl

@[simp] theorem card_slot : Fintype.card Slot = 3 := by decide

abbrev I (u : ℕ) := Fin u → Bool
abbrev J (u : ℕ) := Fin u → ClauseAnswer
abbrev ClauseContext (F : Formula) (u : ℕ) := Fin u → Fin F.clauses.length
abbrev VariableContext (F : Formula) (u : ℕ) := Fin u → Fin F.variables
abbrev SlotContext (u : ℕ) := Fin u → Slot

@[simp] theorem card_I (u : ℕ) : Fintype.card (I u) = 2 ^ u := by simp [I]
@[simp] theorem card_J (u : ℕ) : Fintype.card (J u) = 8 ^ u := by simp [J]
@[simp] theorem card_ClauseContext (F : Formula) (u : ℕ) :
    Fintype.card (ClauseContext F u) = F.clauses.length ^ u := by simp [ClauseContext]
@[simp] theorem card_VariableContext (F : Formula) (u : ℕ) :
    Fintype.card (VariableContext F u) = F.variables ^ u := by simp [VariableContext]
@[simp] theorem card_SlotContext (u : ℕ) :
    Fintype.card (SlotContext u) = 3 ^ u := by simp [SlotContext]

def localConsistent {n : ℕ} (clause : Clause n) (answer : ClauseAnswer) : Bool :=
  decide (∀ s s' : Slot, nameAt clause s = nameAt clause s' →
    answerAt answer s = answerAt answer s')
@[simp] theorem localConsistent_eq_true_iff {n : ℕ}
    (clause : Clause n) (answer : ClauseAnswer) :
    localConsistent clause answer = true ↔
      ∀ s s' : Slot, nameAt clause s = nameAt clause s' →
        answerAt answer s = answerAt answer s' := by
  simp [localConsistent]

def validJ (F : Formula) {u : ℕ} (c : ClauseContext F u) (j : J u) : Bool :=
  decide ((∀ t, localSatisfies (clauseAt F (c t)) (j t) = true) ∧
    ∀ t t' s s', nameAt (clauseAt F (c t)) s = nameAt (clauseAt F (c t')) s' →
      answerAt (j t) s = answerAt (j t') s')

@[simp] theorem validJ_eq_true_iff (F : Formula) {u : ℕ}
    (c : ClauseContext F u) (j : J u) :
    validJ F c j = true ↔
      (∀ t, localSatisfies (clauseAt F (c t)) (j t) = true) ∧
      ∀ t t' s s', nameAt (clauseAt F (c t)) s = nameAt (clauseAt F (c t')) s' →
        answerAt (j t) s = answerAt (j t') s' := by
  simp [validJ]

theorem validJ_satisfies (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (j : J u) (hj : validJ F c j = true) (t : Fin u) :
    localSatisfies (clauseAt F (c t)) (j t) = true :=
  ((validJ_eq_true_iff F c j).mp hj).1 t

theorem validJ_consistent (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (j : J u) (hj : validJ F c j = true) (t : Fin u) (s : Slot)
    (t' : Fin u) (s' : Slot)
    (hname : nameAt (clauseAt F (c t)) s = nameAt (clauseAt F (c t')) s') :
    answerAt (j t) s = answerAt (j t') s' :=
  ((validJ_eq_true_iff F c j).mp hj).2 t t' s s' hname

theorem validJ_localConsistent (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (j : J u) (hj : validJ F c j = true) (t : Fin u) :
    localConsistent (clauseAt F (c t)) (j t) = true := by
  apply (localConsistent_eq_true_iff _ _).mpr
  intro s s' hs
  exact validJ_consistent F c j hj t s t s' hs

def canonicalSlot {n : ℕ} (clause : Clause n) (v : Fin n) : Slot :=
  if nameAt clause .first = v then .first
  else if nameAt clause .second = v then .second else .third

theorem canonicalSlot_name {n : ℕ} (clause : Clause n) (v : Fin n)
    (hv : ∃ s, nameAt clause s = v) : nameAt clause (canonicalSlot clause v) = v := by
  by_cases hfirst : nameAt clause .first = v
  · simp [canonicalSlot, hfirst]
  by_cases hsecond : nameAt clause .second = v
  · simp [canonicalSlot, hfirst, hsecond]
  obtain ⟨s, hs⟩ := hv
  cases s with
  | first => exact False.elim (hfirst hs)
  | second => exact False.elim (hsecond hs)
  | third => simpa [canonicalSlot, hfirst, hsecond] using hs
def pi (F : Formula) {u : ℕ} (c : ClauseContext F u) (v : VariableContext F u)
    (j : J u) : I u :=
  fun t => answerAt (j t) (canonicalSlot (clauseAt F (c t)) (v t))
def sampledVariables (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (s : SlotContext u) : VariableContext F u :=
  fun t => nameAt (clauseAt F (c t)) (s t)
theorem sampledVariables_supported (F : Formula) {u : ℕ}
    (c : ClauseContext F u) (s : SlotContext u) :
    ∀ t, ∃ slot, nameAt (clauseAt F (c t)) slot = sampledVariables F c s t :=
  fun t => ⟨s t, rfl⟩
theorem sampled_eq_pi (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (v : VariableContext F u) (j : J u) (hj : validJ F c j = true)
    (t : Fin u) (s : Slot) (hs : nameAt (clauseAt F (c t)) s = v t) :
    answerAt (j t) s = pi F c v j t := by
  exact validJ_consistent F c j hj t s t
    (canonicalSlot (clauseAt F (c t)) (v t))
    (hs.trans (canonicalSlot_name _ _ ⟨s, hs⟩).symm)
theorem sampled_answers_eq_of_visible_eq (F : Formula) {u : ℕ}
    (c : ClauseContext F u) (s s' : SlotContext u) (j : J u)
    (hj : validJ F c j = true)
    (hvisible : sampledVariables F c s = sampledVariables F c s') :
    (fun t => answerAt (j t) (s t)) = fun t => answerAt (j t) (s' t) := by
  funext t
  exact validJ_consistent F c j hj t (s t) t (s' t) (congrFun hvisible t)
def honestJ (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (assignment : Fin F.variables → Bool) : J u :=
  fun t => honestAnswer (clauseAt F (c t)) assignment
def honestI (F : Formula) {u : ℕ} (v : VariableContext F u)
    (assignment : Fin F.variables → Bool) : I u :=
  fun t => assignment (v t)
theorem honestJ_valid (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (assignment : Fin F.variables → Bool)
    (hs : ∀ clause ∈ F.clauses, clause.eval assignment = true) :
    validJ F c (honestJ F c assignment) = true := by
  apply (validJ_eq_true_iff F c _).mpr
  constructor
  · intro t
    exact (honest_satisfies _ assignment).trans (hs _ (List.getElem_mem _))
  · intro t t' s s' hname
    simp only [honestJ, honest_answerAt]
    exact congrArg assignment hname
theorem pi_honest (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (v : VariableContext F u) (assignment : Fin F.variables → Bool)
    (hsupport : ∀ t, ∃ s, nameAt (clauseAt F (c t)) s = v t) :
    pi F c v (honestJ F c assignment) = honestI F v assignment := by
  funext t
  exact (honest_answerAt _ assignment _).trans
    (congrArg assignment (canonicalSlot_name _ _ (hsupport t)))
theorem pi_honest_sampled (F : Formula) {u : ℕ} (c : ClauseContext F u)
    (s : SlotContext u) (assignment : Fin F.variables → Bool) :
    pi F c (sampledVariables F c s) (honestJ F c assignment) =
      honestI F (sampledVariables F c s) assignment :=
  pi_honest F c _ assignment (sampledVariables_supported F c s)
def baseAccepts (F : Formula) (v : Fin F.variables) (c : Fin F.clauses.length)
    (i : Bool) (j : ClauseAnswer) : Bool :=
  decide (localSatisfies (clauseAt F c) j = true ∧
    localConsistent (clauseAt F c) j = true ∧
    answerAt j (canonicalSlot (clauseAt F c) v) = i)
@[simp] theorem baseAccepts_eq_true_iff (F : Formula) (v : Fin F.variables)
    (c : Fin F.clauses.length) (i : Bool) (j : ClauseAnswer) :
    baseAccepts F v c i j = true ↔
      localSatisfies (clauseAt F c) j = true ∧
      localConsistent (clauseAt F c) j = true ∧
      answerAt j (canonicalSlot (clauseAt F c) v) = i := by
  simp [baseAccepts]
theorem projection_implies_coordinate_accepts (F : Formula) {u : ℕ}
    (c : ClauseContext F u) (v : VariableContext F u) (i : I u) (j : J u)
    (hj : validJ F c j = true) (hpi : pi F c v j = i) (t : Fin u) :
    baseAccepts F (v t) (c t) (i t) (j t) = true := by
  exact (baseAccepts_eq_true_iff F _ _ _ _).mpr
    ⟨validJ_satisfies F c j hj t, validJ_localConsistent F c j hj t, congrFun hpi t⟩
theorem baseAccepts_implies_sampled (F : Formula) (c : Fin F.clauses.length)
    (s : Slot) (i : Bool) (j : ClauseAnswer)
    (h : baseAccepts F (nameAt (clauseAt F c) s) c i j = true) :
    (localSatisfies (clauseAt F c) j && decide (answerAt j s = i)) = true := by
  obtain ⟨hsat, hcons, heq⟩ := (baseAccepts_eq_true_iff F _ _ _ _).mp h
  have hname := canonicalSlot_name (clauseAt F c) (nameAt (clauseAt F c) s) ⟨s, rfl⟩
  have hans := (localConsistent_eq_true_iff _ _).mp hcons s
    (canonicalSlot (clauseAt F c) (nameAt (clauseAt F c) s)) hname.symm
  simp [hsat, hans.trans heq]
theorem projection_implies_sampled_accepts (F : Formula) {u : ℕ}
    (c : ClauseContext F u) (s : SlotContext u) (i : I u) (j : J u)
    (hj : validJ F c j = true) (hpi : pi F c (sampledVariables F c s) j = i) :
    ∀ t, (localSatisfies (clauseAt F (c t)) (j t) &&
      decide (answerAt (j t) (s t) = i t)) = true := by
  intro t
  exact baseAccepts_implies_sampled F (c t) (s t) (i t) (j t)
    (projection_implies_coordinate_accepts F c _ i j hj hpi t)
end BinPackingGames.Foundations.Hastad.SourceContexts
namespace BinPackingGames.Foundations.Hastad
section
open scoped BigOperators
open Finset
open BinPackingGames.Foundations.Games
variable {I J : Type*} [Fintype I] [DecidableEq I]
  [Fintype J] [DecidableEq J]
def supportResponse (fallback : I) (s : Cube I) : FiniteDistribution I where
  weight i := if (support s).Nonempty then
    if i ∈ support s then 1 / ((support s).card : ℝ) else 0
    else if i = fallback then 1 else 0
  nonnegative i := by
    split_ifs <;> positivity
  normalized := by
    by_cases hs : (support s).Nonempty
    · have hc : ((support s).card : ℝ) ≠ 0 :=
        ne_of_gt (Nat.cast_pos.mpr (Finset.card_pos.mpr hs))
      simp [hs, hc]
    · simp [hs]
theorem supportResponse_expectation (fallback : I) (s : Cube I) (H : I → ℝ)
    (hs : (support s).Nonempty) :
    (supportResponse fallback s).expectation H =
      (∑ i ∈ support s, H i) / ((support s).card : ℝ) := by
  simp only [FiniteDistribution.expectation, supportResponse, hs, ite_true]
  simp_rw [ite_mul, zero_mul]
  rw [Finset.sum_ite_mem]
  simp only [div_eq_mul_inv, one_mul, Finset.univ_inter]
  rw [← Finset.mul_sum]
  ring
def fourierResponse (fallback : I) (F : Cube I → Bool) : FiniteDistribution I where
  weight i := ∑ s, coefficient (fun f => bitSign (F f)) s ^ 2 *
    (supportResponse fallback s).weight i
  nonnegative i := Finset.sum_nonneg fun s _ =>
    mul_nonneg (sq_nonneg _) ((supportResponse fallback s).nonnegative i)
  normalized := by
    rw [Finset.sum_comm]
    simp_rw [← Finset.mul_sum, FiniteDistribution.normalized, mul_one]
    exact sign_parseval F
theorem fourierResponse_expectation (fallback : I) (F : Cube I → Bool)
    (H : I → ℝ) :
    (fourierResponse fallback F).expectation H =
      ∑ s, coefficient (fun f => bitSign (F f)) s ^ 2 *
        (supportResponse fallback s).expectation H := by
  unfold FiniteDistribution.expectation fourierResponse
  simp_rw [Finset.sum_mul]
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro s _
  rw [Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro i _
  ring
theorem response_expectation_nonnegative {K : Type*} [Fintype K]
    (law : FiniteDistribution K) (H : K → ℝ) (hH : ∀ k, 0 ≤ H k) :
    0 ≤ law.expectation H :=
  Finset.sum_nonneg fun k _ => mul_nonneg (law.nonnegative k) (hH k)
def responseAgreement (π : J → I) (fallbackI : I) (fallbackJ : J)
    (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  (fourierResponse fallbackI A).expectation fun i =>
    (fourierResponse fallbackJ B).expectation fun j => if π j = i then 1 else 0
def validResponseAgreement (valid : J → Bool) (π : J → I)
    (fallbackI : I) (fallbackJ : J)
    (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  (fourierResponse fallbackI A).expectation fun i =>
    (fourierResponse fallbackJ B).expectation fun j =>
      if π j = i ∧ valid j = true then 1 else 0
theorem supportResponse_validAgreement (valid : J → Bool) (π : J → I)
    (fallbackI : I) (fallbackJ : J) (a : Cube I) (b : Cube J)
    (ha : (support a).Nonempty) (hb : (support b).Nonempty) :
    (supportResponse fallbackI a).expectation (fun i =>
      (supportResponse fallbackJ b).expectation (fun j =>
        if π j = i ∧ valid j = true then 1 else 0)) =
      validAgreementProbability valid π a b := by
  rw [supportResponse_expectation fallbackI a _ ha]
  simp_rw [supportResponse_expectation fallbackJ b _ hb, Finset.sum_boole]
  unfold validAgreementProbability
  simp only [div_eq_mul_inv]
  rw [← Finset.sum_mul]
  ring
theorem validAgreement_le_supportResponses (valid : J → Bool) (π : J → I)
    (fallbackI : I) (fallbackJ : J) (a : Cube I) (b : Cube J) :
    validAgreementProbability valid π a b ≤
      (supportResponse fallbackI a).expectation (fun i =>
        (supportResponse fallbackJ b).expectation (fun j =>
          if π j = i ∧ valid j = true then 1 else 0)) := by
  by_cases ha : (support a).Nonempty
  · by_cases hb : (support b).Nonempty
    · exact le_of_eq (supportResponse_validAgreement valid π fallbackI fallbackJ a b ha hb).symm
    · have he : support b = ∅ := Finset.not_nonempty_iff_eq_empty.mp hb
      simp only [validAgreementProbability, he, Finset.card_empty, Nat.cast_zero,
        mul_zero, div_zero]
      apply response_expectation_nonnegative
      intro i
      apply response_expectation_nonnegative
      intro j
      split_ifs <;> norm_num
  · have he : support a = ∅ := Finset.not_nonempty_iff_eq_empty.mp ha
    simp only [validAgreementProbability, he, Finset.sum_empty, Finset.card_empty,
      Nat.cast_zero, zero_mul, div_zero]
    apply response_expectation_nonnegative
    intro i
    apply response_expectation_nonnegative
    intro j
    split_ifs <;> norm_num
theorem validDecoderSuccess_le_response (valid : J → Bool) (π : J → I)
    (fallbackI : I) (fallbackJ : J) (A : Cube I → Bool) (B : Cube J → Bool) :
    validDecoderSuccess valid π A B ≤
      validResponseAgreement valid π fallbackI fallbackJ A B := by
  unfold validDecoderSuccess validResponseAgreement
  rw [FiniteDistribution.expectation_comm]
  rw [fourierResponse_expectation fallbackJ B]
  apply Finset.sum_le_sum
  intro b _
  apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
  rw [FiniteDistribution.expectation_comm]
  rw [fourierResponse_expectation fallbackI A]
  apply Finset.sum_le_sum
  intro a _
  apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
  exact validAgreement_le_supportResponses valid π fallbackI fallbackJ a b
theorem conditioned_folded_response_bound (ε : ℝ) (π : J → I) (valid : J → Bool)
    (i₀ : I) (tableA : HalfCube i₀ → Bool)
    (j₀ : {j : J // valid j = true}) (tableB : HalfCube j₀ → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * testBias ε π (fun f => bitSign (foldedAnswer i₀ tableA f))
      (fun g => bitSign (conditionedFoldedAnswer valid j₀ tableB g)) ^ 2 ≤
      validResponseAgreement valid π i₀ j₀.val (foldedAnswer i₀ tableA)
        (conditionedFoldedAnswer valid j₀ tableB) :=
  (conditioned_folded_decoder_bound ε π valid i₀ tableA j₀ tableB hε hε').trans
    (validDecoderSuccess_le_response valid π i₀ j₀.val _ _)
end
open scoped BigOperators
open Finset
open BinPackingGames.Foundations.Games
theorem distribution_sq_expectation_le {Ω : Type*} [Fintype Ω]
    (law : FiniteDistribution Ω) (F : Ω → ℝ) :
    law.expectation F ^ 2 ≤ law.expectation (fun x => F x ^ 2) := by
  let m := law.expectation F
  have h := response_expectation_nonnegative law (fun x => (F x - m) ^ 2)
    (fun x => sq_nonneg _)
  have he : law.expectation (fun x => (F x - m) ^ 2) =
      law.expectation (fun x => F x ^ 2) - m ^ 2 := by
    unfold FiniteDistribution.expectation
    calc
      _ = ∑ x, ((law.weight x * F x ^ 2 - 2 * m * (law.weight x * F x)) +
          law.weight x * m ^ 2) := by
        apply Finset.sum_congr rfl
        intro x _
        ring
      _ = _ := by
        rw [Finset.sum_add_distrib, Finset.sum_sub_distrib,
          ← Finset.mul_sum, ← Finset.sum_mul, law.normalized]
        change (law.expectation (fun x => F x ^ 2) - 2 * m * m) + 1 * m ^ 2 = _
        simp only [FiniteDistribution.expectation]
        ring
  rw [he] at h
  dsimp [m] at h
  linarith
variable {Q₁ Q₂ I J : Type*}
  [Fintype Q₁] [DecidableEq Q₁] [Fintype Q₂] [DecidableEq Q₂]
  [Fintype I] [DecidableEq I] [Nonempty I]
  [Fintype J] [DecidableEq J] [Nonempty J]
def projectionGame (questions : FiniteDistribution (Q₁ × Q₂))
    (π : Q₁ → Q₂ → J → I) (valid : Q₂ → J → Bool) : Game Q₁ Q₂ I J where
  questions := questions
  accepts q₁ q₂ i j := decide (π q₁ q₂ j = i ∧ valid q₂ j = true)
def questionBias (ε : ℝ) (π : Q₁ → Q₂ → J → I)
    (A : Q₁ → Cube I → Bool) (B : Q₂ → Cube J → Bool) (q : Q₁ × Q₂) : ℝ :=
  testBias ε (π q.1 q.2) (fun f => bitSign (A q.1 f)) (fun g => bitSign (B q.2 g))
def questionAcceptance (ε : ℝ) (π : Q₁ → Q₂ → J → I)
    (A : Q₁ → Cube I → Bool) (B : Q₂ → Cube J → Bool) (q : Q₁ × Q₂) : ℝ :=
  testAcceptance ε (π q.1 q.2) (A q.1) (B q.2)
omit [DecidableEq Q₁] [DecidableEq Q₂] [Nonempty I] [Nonempty J] in
theorem questionAcceptance_eq (questions : FiniteDistribution (Q₁ × Q₂))
    (ε : ℝ) (π : Q₁ → Q₂ → J → I)
    (A : Q₁ → Cube I → Bool) (B : Q₂ → Cube J → Bool) :
    questions.expectation (questionAcceptance ε π A B) =
      (1 + questions.expectation (questionBias ε π A B)) / 2 := by
  unfold questionAcceptance
  simp_rw [testAcceptance_eq]
  unfold questionBias FiniteDistribution.expectation
  simp only [div_eq_mul_inv, mul_add, mul_one, ← mul_assoc,
    Finset.sum_add_distrib, ← Finset.sum_mul, questions.normalized]
inductive ConditionedOracle (valid : J → Bool) where
  | empty (noLegal : ∀ j, valid j = false)
  | stored (j₀ : {j : J // valid j = true}) (table : HalfCube j₀ → Bool)
def ConditionedOracle.answer {valid : J → Bool} : ConditionedOracle valid → Cube J → Bool
  | .empty _ => fun _ => false
  | .stored j₀ table => conditionedFoldedAnswer valid j₀ table
def ConditionedOracle.fallback {valid : J → Bool} (default : J) :
    ConditionedOracle valid → J
  | .empty _ => default
  | .stored j₀ _ => j₀.val
omit [Nonempty I] [Nonempty J] in
theorem testBias_constant_right (ε : ℝ) (π : J → I) (A : Cube I → ℝ) :
    testBias ε π A (fun _ => 1) = 𝔼 f, A f := by
  unfold testBias
  simp only [mul_one, Fintype.expect_const]
  simp_rw [← Finset.sum_mul, noiseWeight_sum, one_mul]
omit [Nonempty I] [Nonempty J] in
theorem empty_conditioned_bias (ε : ℝ) (π : J → I)
    (i₀ : I) (tableA : HalfCube i₀ → Bool) :
    testBias ε π (fun f => bitSign (foldedAnswer i₀ tableA f))
      (fun _ => bitSign false) = 0 := by
  simp only [bitSign, Bool.false_eq_true, ite_false]
  rw [testBias_constant_right]
  have hz := foldedAnswer_zero_coefficient i₀ tableA
  simpa [coefficient, walsh, bitSign] using hz
omit [Nonempty I] [Nonempty J] in
theorem conditionedOracle_response_bound (ε : ℝ) (π : J → I) (valid : J → Bool)
    (i₀ : I) (tableA : HalfCube i₀ → Bool) (default : J)
    (right : ConditionedOracle valid) (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * testBias ε π (fun f => bitSign (foldedAnswer i₀ tableA f))
      (fun g => bitSign (right.answer g)) ^ 2 ≤
      validResponseAgreement valid π i₀ (right.fallback default)
        (foldedAnswer i₀ tableA) right.answer := by
  cases right with
  | empty noLegal =>
      simp only [ConditionedOracle.answer, ConditionedOracle.fallback,
        empty_conditioned_bias, pow_two, mul_zero]
      unfold validResponseAgreement
      apply response_expectation_nonnegative
      intro i
      apply response_expectation_nonnegative
      intro j
      split_ifs <;> norm_num
  | stored j₀ table =>
      exact conditioned_folded_response_bound ε π valid i₀ tableA j₀ table hε hε'
theorem folded_game_decoder_bound
    (questions : FiniteDistribution (Q₁ × Q₂))
    (ε : ℝ) (π : Q₁ → Q₂ → J → I) (valid : Q₂ → J → Bool)
    (i₀ : Q₁ → I) (tableA : ∀ q, HalfCube (i₀ q) → Bool)
    (j₀ : ∀ q, {j : J // valid q j = true})
    (tableB : ∀ q, HalfCube (j₀ q) → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * questions.expectation (questionBias ε π
      (fun q => foldedAnswer (i₀ q) (tableA q))
      (fun q => conditionedFoldedAnswer (valid q) (j₀ q) (tableB q))) ^ 2 ≤
      (projectionGame questions π valid).value := by
  let A := fun q => foldedAnswer (i₀ q) (tableA q)
  let B := fun q => conditionedFoldedAnswer (valid q) (j₀ q) (tableB q)
  let bias := questionBias ε π A B
  calc
    _ ≤ 4 * ε * questions.expectation (fun q => bias q ^ 2) :=
      mul_le_mul_of_nonneg_left (distribution_sq_expectation_le questions bias) (by linarith)
    _ = questions.expectation (fun q => 4 * ε * bias q ^ 2) := by
      unfold FiniteDistribution.expectation
      rw [Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro q _
      ring
    _ ≤ questions.expectation (fun q =>
        validResponseAgreement (valid q.2) (π q.1 q.2) (i₀ q.1) (j₀ q.2).val
          (A q.1) (B q.2)) := by
      unfold FiniteDistribution.expectation
      apply Finset.sum_le_sum
      intro q _
      apply mul_le_mul_of_nonneg_left _ (questions.nonnegative q)
      exact conditioned_folded_response_bound ε (π q.1 q.2) (valid q.2)
        (i₀ q.1) (tableA q.1) (j₀ q.2) (tableB q.2) hε hε'
    _ = (projectionGame questions π valid).stochasticSuccess
        (fun q => fourierResponse (i₀ q) (A q))
        (fun q => fourierResponse (j₀ q).val (B q)) := by
      simp [Game.stochasticSuccess, projectionGame, validResponseAgreement]
    _ ≤ _ := Game.stochasticSuccess_le_value _ _ _
theorem conditionedOracle_game_decoder_bound
    (questions : FiniteDistribution (Q₁ × Q₂))
    (ε : ℝ) (π : Q₁ → Q₂ → J → I) (valid : Q₂ → J → Bool)
    (i₀ : Q₁ → I) (tableA : ∀ q, HalfCube (i₀ q) → Bool)
    (default : J) (right : ∀ q, ConditionedOracle (valid q))
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * questions.expectation (questionBias ε π
      (fun q => foldedAnswer (i₀ q) (tableA q)) (fun q => (right q).answer)) ^ 2 ≤
      (projectionGame questions π valid).value := by
  let A := fun q => foldedAnswer (i₀ q) (tableA q)
  let B := fun q => (right q).answer
  let bias := questionBias ε π A B
  calc
    _ ≤ 4 * ε * questions.expectation (fun q => bias q ^ 2) :=
      mul_le_mul_of_nonneg_left (distribution_sq_expectation_le questions bias) (by linarith)
    _ = questions.expectation (fun q => 4 * ε * bias q ^ 2) := by
      unfold FiniteDistribution.expectation
      rw [Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro q _
      ring
    _ ≤ questions.expectation (fun q =>
        validResponseAgreement (valid q.2) (π q.1 q.2) (i₀ q.1)
          ((right q.2).fallback default) (A q.1) (B q.2)) := by
      unfold FiniteDistribution.expectation
      apply Finset.sum_le_sum
      intro q _
      apply mul_le_mul_of_nonneg_left _ (questions.nonnegative q)
      exact conditionedOracle_response_bound ε (π q.1 q.2) (valid q.2)
        (i₀ q.1) (tableA q.1) default (right q.2) hε hε'
    _ = (projectionGame questions π valid).stochasticSuccess
        (fun q => fourierResponse (i₀ q) (A q))
        (fun q => fourierResponse ((right q).fallback default) (B q)) := by
      simp [Game.stochasticSuccess, projectionGame, validResponseAgreement]
    _ ≤ _ := Game.stochasticSuccess_le_value _ _ _
theorem conditionedOracle_acceptance_bound
    (questions : FiniteDistribution (Q₁ × Q₂))
    (ε δ : ℝ) (π : Q₁ → Q₂ → J → I) (valid : Q₂ → J → Bool)
    (i₀ : Q₁ → I) (tableA : ∀ q, HalfCube (i₀ q) → Bool)
    (default : J) (right : ∀ q, ConditionedOracle (valid q))
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) (hδ : 0 ≤ δ)
    (hvalue : (projectionGame questions π valid).value ≤ 4 * ε * δ ^ 2) :
    questions.expectation (questionAcceptance ε π
      (fun q => foldedAnswer (i₀ q) (tableA q)) (fun q => (right q).answer)) ≤
      (1 + δ) / 2 := by
  have hd := conditionedOracle_game_decoder_bound questions ε π valid i₀ tableA
    default right hε hε'
  have hpos : 0 < 4 * ε := by positivity
  have hs := le_of_mul_le_mul_left (hd.trans hvalue) hpos
  rw [questionAcceptance_eq]
  nlinarith
end BinPackingGames.Foundations.Hastad
end
end OAI