lean/OAI/Computability/BinPacking/PCP/Queries.lean

import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Basic.Real.Basic
import Mathlib.Logic.Equiv.Prod
import OAI.Computability.BinPacking.Information.GameDecoder
import OAI.Computability.BinPacking.PCP.AlphabetRetraction

namespace OAI

noncomputable section

namespace BinPackingGames.Foundations.PCP.UniformRestriction

open scoped BigOperators

variable {B : Type*} (p : B → Prop)

def restrict (f : B → Bool) : {b // p b} → Bool := fun b => f b.1

variable [DecidablePred p]

def extend (g : {b // p b} → Bool) : B → Bool :=
  fun b => if hb : p b then g ⟨b, hb⟩ else false

def canonicalize (f : B → Bool) : B → Bool := extend p (restrict p f)

omit [DecidablePred p] in
@[simp] theorem restrict_apply (f : B → Bool) (b : {b // p b}) :
    restrict p f b = f b.1 := rfl

@[simp] theorem extend_apply_of_mem (g : {b // p b} → Bool) (b : B) (hb : p b) :
    extend p g b = g ⟨b, hb⟩ := by
  simp [extend, hb]

@[simp] theorem extend_apply_of_not_mem (g : {b // p b} → Bool) (b : B)
    (hb : ¬ p b) : extend p g b = false := by
  simp [extend, hb]

@[simp] theorem restrict_extend (g : {b // p b} → Bool) :
    restrict p (extend p g) = g := by
  funext b
  simp [restrict, extend, b.property]

@[simp] theorem canonicalize_apply_of_mem (f : B → Bool) (b : B) (hb : p b) :
    canonicalize p f b = f b := by
  simp [canonicalize, hb]

@[simp] theorem canonicalize_apply_of_not_mem (f : B → Bool) (b : B)
    (hb : ¬ p b) : canonicalize p f b = false := by
  simp [canonicalize, hb]

@[simp] theorem restrict_canonicalize (f : B → Bool) :
    restrict p (canonicalize p f) = restrict p f := by
  exact restrict_extend p (restrict p f)

@[simp] theorem canonicalize_idempotent (f : B → Bool) :
    canonicalize p (canonicalize p f) = canonicalize p f := by
  simp only [canonicalize, restrict_extend]

theorem canonicalize_eq_iff (f g : B → Bool) :
    canonicalize p f = canonicalize p g ↔ restrict p f = restrict p g := by
  constructor
  · intro h
    simpa only [restrict_canonicalize] using congrArg (restrict p) h
  · intro h
    exact congrArg (extend p) h

variable [Fintype B] [DecidableEq B]

theorem expect_restrict (h : ({b // p b} → Bool) → ℝ) :
    (𝔼 f : B → Bool, h (restrict p f)) = 𝔼 g : {b // p b} → Bool, h g := by
  calc
    (𝔼 f : B → Bool, h (restrict p f)) =
        𝔼 g : ({b // p b} → Bool) × ({b // ¬ p b} → Bool), h g.1 :=
      Fintype.expect_equiv (Equiv.piEquivPiSubtypeProd p (fun _ => Bool))
        _ _ (fun _ => rfl)
    _ = 𝔼 g : {b // p b} → Bool, 𝔼 _k : {b // ¬ p b} → Bool, h g := by
      simpa only [Finset.univ_product_univ] using
        Finset.expect_product
          (Finset.univ : Finset ({b // p b} → Bool))
          (Finset.univ : Finset ({b // ¬ p b} → Bool))
          (fun g : ({b // p b} → Bool) × ({b // ¬ p b} → Bool) => h g.1)
    _ = 𝔼 g : {b // p b} → Bool, h g := by
      simp only [Fintype.expect_const]

theorem expect_canonicalize (h : (B → Bool) → ℝ) :
    (𝔼 f : B → Bool, h (canonicalize p f)) =
      𝔼 g : {b // p b} → Bool, h (extend p g) :=
  expect_restrict p (fun g => h (extend p g))

theorem expect_restrict_pair (h : ({b // p b} → Bool) → ({b // p b} → Bool) → ℝ) :
    (𝔼 f : B → Bool, 𝔼 g : B → Bool, h (restrict p f) (restrict p g)) =
      𝔼 f : {b // p b} → Bool, 𝔼 g : {b // p b} → Bool, h f g := by
  calc
    _ = 𝔼 f : B → Bool, 𝔼 g : {b // p b} → Bool, h (restrict p f) g := by
      apply Finset.expect_congr rfl
      intro f _
      exact expect_restrict p (h (restrict p f))
    _ = _ := expect_restrict p (fun f => 𝔼 g : {b // p b} → Bool, h f g)

end BinPackingGames.Foundations.PCP.UniformRestriction

namespace BinPackingGames.Foundations.PCP.CodeComposition

open scoped BigOperators
open BinPackingGames.Foundations.Hastad

def disagreement (a b : Bool) : ℝ := if a = b then 0 else 1

@[simp] theorem disagreement_self (a : Bool) : disagreement a a = 0 := by
  simp [disagreement]

theorem disagreement_nonnegative (a b : Bool) : 0 ≤ disagreement a b := by
  cases a <;> cases b <;> norm_num [disagreement]

theorem disagreement_le_one (a b : Bool) : disagreement a b ≤ 1 := by
  cases a <;> cases b <;> norm_num [disagreement]

theorem disagreement_comm (a b : Bool) : disagreement a b = disagreement b a := by
  simp [disagreement, eq_comm]

theorem disagreement_triangle (a b c : Bool) :
    disagreement a c ≤ disagreement a b + disagreement b c := by
  cases a <;> cases b <;> cases c <;> norm_num [disagreement]

theorem disagreement_eq_sign (a b : Bool) :
    disagreement a b = (1 - bitSign a * bitSign b) / 2 := by
  cases a <;> cases b <;> norm_num [disagreement, bitSign]

def relativeDistance {X : Type*} [Fintype X] (f g : X → Bool) : ℝ :=
  𝔼 x, disagreement (f x) (g x)

@[simp] theorem relativeDistance_self {X : Type*} [Fintype X] (f : X → Bool) :
    relativeDistance f f = 0 := by
  simp [relativeDistance]

theorem relativeDistance_nonnegative {X : Type*} [Fintype X] (f g : X → Bool) :
    0 ≤ relativeDistance f g :=
  Finset.expect_nonneg (fun _ _ => disagreement_nonnegative _ _)

theorem relativeDistance_comm {X : Type*} [Fintype X] (f g : X → Bool) :
    relativeDistance f g = relativeDistance g f := by
  simp only [relativeDistance, disagreement_comm]

theorem relativeDistance_triangle {X : Type*} [Fintype X] (f g h : X → Bool) :
    relativeDistance f h ≤ relativeDistance f g + relativeDistance g h := by
  rw [relativeDistance, relativeDistance, relativeDistance,
    ← Finset.expect_add_distrib]
  exact Finset.expect_le_expect (fun _ _ => disagreement_triangle _ _ _)

theorem relativeDistance_eq_sign {X : Type*} [Fintype X] [Nonempty X]
    (f g : X → Bool) :
    relativeDistance f g = (1 - 𝔼 x, bitSign (f x) * bitSign (g x)) / 2 := by
  simp only [relativeDistance, disagreement_eq_sign]
  rw [← Finset.expect_div, Finset.expect_sub_distrib]
  simp

def codeword {A : Type*} (a : A) : Cube A → Bool := fun x => x a

private def singletonMask {A : Type*} [DecidableEq A] (a : A) : Cube A :=
  fun i => decide (i = a)

private theorem walsh_singletonMask {A : Type*} [Fintype A] [DecidableEq A]
    (a : A) (x : Cube A) : walsh (singletonMask a) x = bitSign (x a) := by
  classical
  unfold walsh singletonMask
  rw [Finset.prod_eq_single a]
  · simp
  · intro b _ hba
    simp [hba, bitSign]
  · simp

private theorem singletonMask_ne {A : Type*} [DecidableEq A] {a b : A} (hab : a ≠ b) :
    singletonMask a ≠ singletonMask b := by
  intro h
  have he := congrFun h a
  simp [singletonMask, hab] at he

theorem codeword_distance {A : Type*} [Fintype A] [DecidableEq A]
    {a b : A} (hab : a ≠ b) :
    relativeDistance (codeword a) (codeword b) = 1 / 2 := by
  rw [relativeDistance_eq_sign]
  have horth := walsh_orthogonality (singletonMask a) (singletonMask b)
  rw [ite_eq_right (singletonMask_ne hab)] at horth
  simp_rw [walsh_singletonMask] at horth
  change (1 - 𝔼 x : Cube A, bitSign (x a) * bitSign (x b)) / 2 = 1 / 2
  rw [horth]
  norm_num

theorem exists_nearest {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]
    (f : Cube A → Bool) :
    ∃ a : A, ∀ b : A, relativeDistance f (codeword a) ≤ relativeDistance f (codeword b) := by
  classical
  obtain ⟨a, _, ha⟩ := Finset.exists_min_image Finset.univ
    (fun a : A => relativeDistance f (codeword a)) Finset.univ_nonempty
  exact ⟨a, fun b => ha b (Finset.mem_univ b)⟩

def nearest {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A] (f : Cube A → Bool) : A :=
  Classical.choose (exists_nearest f)

theorem nearest_minimal {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]
    (f : Cube A → Bool) (a : A) :
    relativeDistance f (codeword (nearest f)) ≤ relativeDistance f (codeword a) :=
  Classical.choose_spec (exists_nearest f) a

theorem distance_from_other_codeword {A : Type*} [Fintype A] [DecidableEq A]
    [Nonempty A] (f : Cube A → Bool) {a : A} (ha : a ≠ nearest f) :
    1 / 4 ≤ relativeDistance f (codeword a) := by
  have htri := relativeDistance_triangle (codeword (nearest f)) f (codeword a)
  rw [codeword_distance (Ne.symm ha), relativeDistance_comm (codeword (nearest f)) f] at htri
  have hmin := nearest_minimal f a
  linarith

def pairWord {X : Type*} (left right : X → Bool) : Bool × X → Bool :=
  fun p => if p.1 then right p.2 else left p.2

theorem pairWord_distance {X : Type*} [Fintype X] [Nonempty X]
    (left right left' right' : X → Bool) :
    relativeDistance (pairWord left right) (pairWord left' right') =
      (relativeDistance left left' + relativeDistance right right') / 2 := by
  classical
  simp [relativeDistance, Fintype.expect_eq_sum_div_card, Fintype.card_prod,
    Fintype.sum_prod_type, pairWord]
  ring

theorem rejected_pair_far {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]
    (accepts : A → A → Bool) (left right : Cube A → Bool)
    (hbad : accepts (nearest left) (nearest right) = false)
    (a b : A) (hgood : accepts a b = true) :
    1 / 8 ≤ relativeDistance (pairWord left right)
      (pairWord (codeword a) (codeword b)) := by
  rw [pairWord_distance]
  have hleft := relativeDistance_nonnegative left (codeword a)
  have hright := relativeDistance_nonnegative right (codeword b)
  by_cases ha : a = nearest left
  · have hb : b ≠ nearest right := by
      intro hb
      rw [ha, hb, hbad] at hgood
      cases hgood
    have hsep := distance_from_other_codeword right hb
    linarith
  · have hsep := distance_from_other_codeword left ha
    linarith

end BinPackingGames.Foundations.PCP.CodeComposition

namespace BinPackingGames.Foundations.PCP.AssignmentTester

open BinPackingGames.Foundations.Hastad
open scoped BigOperators
open Finset

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

def truth (b : Bool) : ℝ := if b then 1 else 0

def disagree (a b : Bool) : ℝ := if a = b then 0 else 1

theorem truth_nonneg (b : Bool) : 0 ≤ truth b := by cases b <;> norm_num [truth]

theorem disagree_nonneg (a b : Bool) : 0 ≤ disagree a b := by
  cases a <;> cases b <;> norm_num [disagree]

theorem disagree_eq (a b : Bool) :
    disagree a b = (1 - bitSign a * bitSign b) / 2 := by
  cases a <;> cases b <;> norm_num [disagree, bitSign]

theorem disagree_xor_le (a b c d : Bool) :
    disagree (a ^^ b) (c ^^ d) ≤ disagree a c + disagree b d := by
  cases a <;> cases b <;> cases c <;> cases d <;> norm_num [disagree]

def parity (s f : Cube A) : Bool :=
  decide (Odd ((Finset.univ.filter fun a => s a && f a = true).card))

theorem bitSign_parity (s f : Cube A) : bitSign (parity s f) = walsh s f := by
  have hw : walsh s f = (-1 : ℝ) ^
      (Finset.univ.filter fun a => s a && f a = true).card := by
    simp [walsh, bitSign, Finset.prod_ite]
  rw [hw]
  simpa [parity, bitSign] using
    (sign_power_parity true (Finset.univ.filter fun a => s a && f a = true).card).symm

theorem bitSign_injective : Function.Injective bitSign := by
  intro a b h
  cases a <;> cases b <;> simp_all [bitSign] <;> linarith

theorem parity_xor (s f g : Cube A) :
    parity s (cubeXor f g) = (parity s f ^^ parity s g) := by
  apply bitSign_injective
  rw [bitSign_parity, bitSign_xor, bitSign_parity, bitSign_parity, walsh_xor]

omit [Fintype A] [DecidableEq A] in
theorem cubeXor_comm (f g : Cube A) : cubeXor f g = cubeXor g f := by
  funext a
  simp [cubeXor, Bool.xor_comm]

omit [Fintype A] [DecidableEq A] in
@[simp] theorem cubeXor_right_twice (f g : Cube A) :
    cubeXor (cubeXor f g) g = f := by
  funext a
  change ((f a ^^ g a) ^^ g a) = f a
  cases f a <;> cases g a <;> rfl

theorem expect_cubeXor (F : Cube A → ℝ) (g : Cube A) :
    (𝔼 f, F (cubeXor f g)) = 𝔼 f, F f := by
  rw [Fintype.expect_eq_sum_div_card, Fintype.expect_eq_sum_div_card]
  congr 1
  refine Finset.sum_bij (fun f _ => cubeXor f g) ?_ ?_ ?_ ?_
  · intro f _
    exact Finset.mem_univ _
  · intro f _ h _ heq
    have hh := congrArg (fun x => cubeXor x g) heq
    simpa only [cubeXor_right_twice] using hh
  · intro f _
    exact ⟨cubeXor f g, Finset.mem_univ _, cubeXor_right_twice f g⟩
  · intro f _
    rfl

def distance {K : Type*} [Fintype K] (f g : K → Bool) : ℝ :=
  𝔼 k, disagree (f k) (g k)

theorem distance_nonneg {K : Type*} [Fintype K] (f g : K → Bool) :
    0 ≤ distance f g := by
  apply Finset.expect_nonneg
  intro k _
  exact disagree_nonneg _ _

theorem distance_parity (F : Cube A → Bool) (s : Cube A) :
    distance F (parity s) = (1 - coefficient (fun f => bitSign (F f)) s) / 2 := by
  unfold distance
  simp_rw [disagree_eq, bitSign_parity, div_eq_mul_inv,
    ← Finset.expect_mul, Finset.expect_sub_distrib]
  simp only [Fintype.expect_const]
  rfl

def blrReject (F : Cube A → Bool) : ℝ :=
  𝔼 f, 𝔼 g, disagree (F f ^^ F g) (F (cubeXor f g))

theorem blrReject_nonneg (F : Cube A → Bool) : 0 ≤ blrReject F := by
  apply Finset.expect_nonneg
  intro f _
  apply Finset.expect_nonneg
  intro g _
  exact disagree_nonneg _ _

theorem blr_correlation (F : Cube A → Bool) :
    (𝔼 f, 𝔼 g, bitSign (F f) * bitSign (F g) * bitSign (F (cubeXor f g))) =
      ∑ s, coefficient (fun f => bitSign (F f)) s ^ 3 := by
  simp_rw [mul_assoc, ← Finset.mul_expect]
  conv_lhs =>
    enter [2, f, 2, 2, g, 2, 1]
    rw [cubeXor_comm]
  simp_rw [walsh_autocorrelation, Finset.mul_sum]
  rw [Finset.expect_sum_comm]
  apply Finset.sum_congr rfl
  intro s _
  have heq (f : Cube A) : bitSign (F f) *
      (coefficient (fun f => bitSign (F f)) s ^ 2 * walsh s f) =
    coefficient (fun f => bitSign (F f)) s ^ 2 * (bitSign (F f) * walsh s f) := by ring
  simp_rw [heq, ← Finset.mul_expect]
  change coefficient (fun f => bitSign (F f)) s ^ 2 *
    coefficient (fun f => bitSign (F f)) s = _
  ring

theorem blrReject_fourier (F : Cube A → Bool) :
    blrReject F = (1 - ∑ s, coefficient (fun f => bitSign (F f)) s ^ 3) / 2 := by
  unfold blrReject
  simp_rw [disagree_eq, bitSign_xor, div_eq_mul_inv,
    ← Finset.expect_mul, Finset.expect_sub_distrib]
  simp only [Fintype.expect_const]
  rw [blr_correlation]

theorem blr_near_parity (F : Cube A → Bool) :
    ∃ s : Cube A, distance F (parity s) ≤ blrReject F := by
  classical
  let c := coefficient (fun f => bitSign (F f))
  have hmass : (∑ s, c s ^ 2) = 1 := sign_parseval F
  have hlarge : ∃ s, (∑ t, c t ^ 3) ≤ c s := by
    by_contra hn
    push Not at hn
    have hpos : ∃ s, 0 < c s ^ 2 := by
      by_contra hh
      push Not at hh
      have hz := Finset.sum_nonpos (s := Finset.univ) (fun s _ => hh s)
      linarith
    obtain ⟨s₀, hs₀⟩ := hpos
    have hle (s : Cube A) (_ : s ∈ Finset.univ) :
        c s ^ 3 ≤ (∑ t, c t ^ 3) * c s ^ 2 := by
      have h := mul_le_mul_of_nonneg_right (hn s).le (sq_nonneg (c s))
      nlinarith
    have hlt : c s₀ ^ 3 < (∑ t, c t ^ 3) * c s₀ ^ 2 := by
      have h := mul_lt_mul_of_pos_right (hn s₀) hs₀
      nlinarith
    have hsum := Finset.sum_lt_sum hle ⟨s₀, Finset.mem_univ _, hlt⟩
    rw [← Finset.mul_sum, hmass, mul_one] at hsum
    exact (lt_irrefl _ hsum)
  obtain ⟨s, hs⟩ := hlarge
  refine ⟨s, ?_⟩
  rw [distance_parity, blrReject_fourier]
  dsimp [c] at hs
  linarith

def selfCorrect (F : Cube A → Bool) (x r : Cube A) : Bool :=
  F r ^^ F (cubeXor r x)

theorem parity_selfCorrect (s x r : Cube A) :
    (parity s r ^^ parity s (cubeXor r x)) = parity s x := by
  rw [parity_xor]
  cases parity s r <;> cases parity s x <;> rfl

theorem selfCorrect_error (F : Cube A → Bool) (s x : Cube A) :
    (𝔼 r, disagree (selfCorrect F x r) (parity s x)) ≤ 2 * distance F (parity s) := by
  have hle (r : Cube A) :
      disagree (selfCorrect F x r) (parity s x) ≤
      disagree (F r) (parity s r) +
        disagree (F (cubeXor r x)) (parity s (cubeXor r x)) := by
    rw [← parity_selfCorrect s x r]
    exact disagree_xor_le _ _ _ _
  have h := Finset.expect_le_expect (s := Finset.univ) (fun r _ => hle r)
  rw [Finset.expect_add_distrib,
    expect_cubeXor (fun r => disagree (F r) (parity s r)) x] at h
  change _ ≤ distance F (parity s) + distance F (parity s) at h
  linarith

def singleMask (i : A) : Cube A := fun j => decide (j = i)

def flipAt (i : A) (f : Cube A) : Cube A :=
  fun j => if j = i then !(f j) else f j

omit [Fintype A] in
@[simp] theorem flipAt_twice (i : A) (f : Cube A) : flipAt i (flipAt i f) = f := by
  funext j
  by_cases h : j = i <;> simp [flipAt, h]

omit [Fintype A] in
theorem flipAt_eq_xor (i : A) (f : Cube A) : flipAt i f = cubeXor f (singleMask i) := by
  funext j
  by_cases h : j = i <;> simp [flipAt, cubeXor, singleMask, h]

theorem expect_flipAt (F : Cube A → ℝ) (i : A) :
    (𝔼 f, F (flipAt i f)) = 𝔼 f, F f := by
  simp_rw [flipAt_eq_xor]
  exact expect_cubeXor F _

theorem walsh_singleMask (s : Cube A) (i : A) :
    walsh s (singleMask i) = bitSign (s i) := by
  unfold walsh
  rw [Finset.prod_eq_single i]
  · simp [singleMask]
  · intro j _ hj
    simp [singleMask, hj, bitSign]
  · intro h
    exact False.elim (h (Finset.mem_univ _))

theorem parity_singleMask (s : Cube A) (i : A) : parity s (singleMask i) = s i := by
  apply bitSign_injective
  rw [bitSign_parity, walsh_singleMask]

theorem parity_flipAt (s : Cube A) (i : A) (f : Cube A) (hs : s i = true) :
    parity s (flipAt i f) = !(parity s f) := by
  rw [flipAt_eq_xor, parity_xor, parity_singleMask, hs]
  cases parity s f <;> rfl

def cubeAnd (f g : Cube A) : Cube A := fun a => f a && g a

def andLawFailure (s f g : Cube A) : Bool :=
  parity s (cubeAnd f g) ^^ (parity s f && parity s g)

theorem andLawFailure_flipAt_second (s : Cube A) (i : A) (f g : Cube A)
    (hi : s i = true) :
    andLawFailure s f (flipAt i g) =
      (andLawFailure s f g ^^ (parity s f ^^ f i)) := by
  have hAnd : cubeAnd f (flipAt i g) =
      (if f i then flipAt i (cubeAnd f g) else cubeAnd f g) := by
    funext a
    by_cases ha : a = i
    · subst a
      cases hfi : f i <;> simp [cubeAnd, flipAt, hfi]
    · cases f i <;> simp [cubeAnd, flipAt, ha]
  have hg := parity_flipAt s i g hi
  have hfg := parity_flipAt s i (cubeAnd f g) hi
  cases hfi : f i <;> cases hf : parity s f <;>
    cases hg₀ : parity s g <;> cases ha : parity s (cubeAnd f g) <;>
    simp [andLawFailure, hAnd, hg, hfg, hfi, hf, hg₀, ha]

private theorem four_truth_lower (e₀ e₁ d : Bool) :
    (1 : ℝ) ≤ truth e₀ + truth (e₀ ^^ d) + truth e₁ + truth (e₁ ^^ (!d)) := by
  cases e₀ <;> cases e₁ <;> cases d <;> norm_num [truth]

theorem parity_and_rejection (s : Cube A) (i j : A)
    (hi : s i = true) (hj : s j = true) (hij : i ≠ j) :
    (1 : ℝ) / 4 ≤ 𝔼 f, 𝔼 g, truth (andLawFailure s f g) := by
  have hpoint (f g : Cube A) : (1 : ℝ) ≤
      truth (andLawFailure s f g) + truth (andLawFailure s f (flipAt i g)) +
      truth (andLawFailure s (flipAt j f) g) +
      truth (andLawFailure s (flipAt j f) (flipAt i g)) := by
    rw [andLawFailure_flipAt_second s i f g hi,
      andLawFailure_flipAt_second s i (flipAt j f) g hi]
    have hD : (parity s (flipAt j f) ^^ (flipAt j f) i) =
        !(parity s f ^^ f i) := by
      rw [parity_flipAt s j f hj]
      simp only [flipAt, ite_eq_right hij]
      cases parity s f <;> cases f i <;> simp
    rw [hD]
    exact four_truth_lower _ _ _
  have havg := Finset.expect_le_expect (s := Finset.univ) (fun f _ =>
    Finset.expect_le_expect (s := Finset.univ) (fun g _ => hpoint f g))
  simp only [Finset.expect_add_distrib, Fintype.expect_const] at havg
  have hinner (f : Cube A) :
      (𝔼 g, truth (andLawFailure s f (flipAt i g))) =
        𝔼 g, truth (andLawFailure s f g) :=
    expect_flipAt (fun g => truth (andLawFailure s f g)) i
  have houter : (𝔼 f, 𝔼 g, truth (andLawFailure s (flipAt j f) g)) =
      𝔼 f, 𝔼 g, truth (andLawFailure s f g) :=
    expect_flipAt (fun f => 𝔼 g, truth (andLawFailure s f g)) j
  simp only [hinner, houter] at havg
  linarith

theorem truth_xor (a b : Bool) : truth (a ^^ b) = disagree a b := by
  cases a <;> cases b <;> norm_num [truth, disagree]

theorem disagree_triangle (a b c : Bool) :
    disagree a c ≤ disagree a b + disagree b c := by
  cases a <;> cases b <;> cases c <;> norm_num [disagree]

theorem and_disagree_transfer (p₀ p₁ p₂ q₀ q₁ q₂ : Bool) :
    disagree p₀ (p₁ && p₂) ≤ disagree q₀ (q₁ && q₂) +
      disagree q₀ p₀ + disagree q₁ p₁ + disagree q₂ p₂ := by
  cases p₀ <;> cases p₁ <;> cases p₂ <;>
    cases q₀ <;> cases q₁ <;> cases q₂ <;> norm_num [disagree]

def andReject (F : Cube A → Bool) : ℝ :=
  𝔼 f, 𝔼 g, 𝔼 r₀, 𝔼 r₁, 𝔼 r₂,
    disagree (selfCorrect F (cubeAnd f g) r₀)
      (selfCorrect F f r₁ && selfCorrect F g r₂)

def unitReject (F : Cube A → Bool) : ℝ :=
  𝔼 r, disagree (selfCorrect F (fun _ => true) r) true

variable {K : Type*} [Fintype K] [Nonempty K]

def inputReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) : ℝ :=
  𝔼 k, 𝔼 r, disagree (σ k) (selfCorrect F (fun a => label a k) r)

def testerReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) : ℝ :=
  (blrReject F + andReject F + unitReject F + inputReject label σ F) / 4

theorem andReject_nonneg (F : Cube A → Bool) : 0 ≤ andReject F := by
  apply Finset.expect_nonneg
  intro f _
  apply Finset.expect_nonneg
  intro g _
  apply Finset.expect_nonneg
  intro r₀ _
  apply Finset.expect_nonneg
  intro r₁ _
  apply Finset.expect_nonneg
  intro r₂ _
  exact disagree_nonneg _ _

theorem unitReject_nonneg (F : Cube A → Bool) : 0 ≤ unitReject F := by
  apply Finset.expect_nonneg
  intro r _
  exact disagree_nonneg _ _

omit [Nonempty K] in
theorem inputReject_nonneg (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) :
    0 ≤ inputReject label σ F := by
  apply Finset.expect_nonneg
  intro k _
  apply Finset.expect_nonneg
  intro r _
  exact disagree_nonneg _ _

theorem andReject_transfer (F : Cube A → Bool) (s : Cube A) :
    (𝔼 f, 𝔼 g, truth (andLawFailure s f g)) ≤
      andReject F + 6 * distance F (parity s) := by
  have hlocal (f g : Cube A) : truth (andLawFailure s f g) ≤
      (𝔼 r₀, 𝔼 r₁, 𝔼 r₂,
        disagree (selfCorrect F (cubeAnd f g) r₀)
          (selfCorrect F f r₁ && selfCorrect F g r₂)) +
      6 * distance F (parity s) := by
    have hp (r₀ r₁ r₂ : Cube A) := and_disagree_transfer
      (parity s (cubeAnd f g)) (parity s f) (parity s g)
      (selfCorrect F (cubeAnd f g) r₀) (selfCorrect F f r₁) (selfCorrect F g r₂)
    have havg := Finset.expect_le_expect (s := Finset.univ) (fun r₀ _ =>
      Finset.expect_le_expect (s := Finset.univ) (fun r₁ _ =>
        Finset.expect_le_expect (s := Finset.univ) (fun r₂ _ => hp r₀ r₁ r₂)))
    simp only [Finset.expect_add_distrib, Fintype.expect_const] at havg
    have h₀ := selfCorrect_error F s (cubeAnd f g)
    have h₁ := selfCorrect_error F s f
    have h₂ := selfCorrect_error F s g
    rw [andLawFailure, truth_xor]
    linarith
  have h := Finset.expect_le_expect (s := Finset.univ) (fun f _ =>
    Finset.expect_le_expect (s := Finset.univ) (fun g _ => hlocal f g))
  simpa only [Finset.expect_add_distrib, Fintype.expect_const, andReject] using h

theorem parity_zero (f : Cube A) : parity (fun _ => false) f = false := by
  apply bitSign_injective
  rw [bitSign_parity]
  simp [walsh, bitSign]

theorem unitReject_zero_parity (F : Cube A → Bool) :
    1 - 2 * distance F (parity (fun _ => false)) ≤ unitReject F := by
  have herr := selfCorrect_error F (fun _ => false) (fun _ => true)
  simp only [parity_zero] at herr
  have heq : unitReject F +
      (𝔼 r, disagree (selfCorrect F (fun _ => true) r) false) = 1 := by
    unfold unitReject
    rw [← Finset.expect_add_distrib]
    have hp (b : Bool) : disagree b true + disagree b false = 1 := by
      cases b <;> norm_num [disagree]
    simp only [hp, Fintype.expect_const]
  linarith

theorem parity_singleMask_eval (a : A) (f : Cube A) :
    parity (singleMask a) f = f a := by
  apply bitSign_injective
  rw [bitSign_parity, walsh_symm, walsh_singleMask]

theorem inputReject_transfer (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) (a : A) :
    distance σ (label a) ≤ inputReject label σ F +
      2 * distance F (parity (singleMask a)) := by
  have hlocal (k : K) : disagree (σ k) (label a k) ≤
      (𝔼 r, disagree (σ k) (selfCorrect F (fun a => label a k) r)) +
        2 * distance F (parity (singleMask a)) := by
    have hp (r : Cube A) := disagree_triangle (σ k)
      (selfCorrect F (fun a => label a k) r) (label a k)
    have havg := Finset.expect_le_expect (s := Finset.univ) (fun r _ => hp r)
    simp only [Fintype.expect_const, Finset.expect_add_distrib] at havg
    have herr := selfCorrect_error F (singleMask a) (fun a => label a k)
    simp only [parity_singleMask_eval] at herr
    linarith
  have h := Finset.expect_le_expect (s := Finset.univ) (fun k _ => hlocal k)
  simpa only [Finset.expect_add_distrib, Fintype.expect_const, inputReject, distance] using h

theorem tester_proximity_soundness (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) {δ : ℝ} (hδ : 0 < δ) (hδ' : δ ≤ 1 / 4)
    (hfar : ∀ a, δ ≤ distance σ (label a)) :
    δ / 256 ≤ testerReject label σ F := by
  have hblr := blrReject_nonneg F
  have hand := andReject_nonneg F
  have hunit := unitReject_nonneg F
  have hin := inputReject_nonneg label σ F
  by_cases hb : δ / 64 ≤ blrReject F
  · unfold testerReject
    linarith
  obtain ⟨s, hs⟩ := blr_near_parity F
  by_cases hz : s = fun _ => false
  · subst s
    have hu := unitReject_zero_parity F
    unfold testerReject
    linarith
  have hsome : ∃ i, s i = true := by
    by_contra hh
    push Not at hh
    apply hz
    funext i
    have h := hh i
    cases hi : s i <;> simp_all
  obtain ⟨i, hi⟩ := hsome
  by_cases htwo : ∃ j, i ≠ j ∧ s j = true
  · obtain ⟨j, hij, hj⟩ := htwo
    have ha := parity_and_rejection s i j hi hj hij
    have ht := andReject_transfer F s
    unfold testerReject
    linarith
  · have heq : s = singleMask i := by
      funext j
      by_cases hji : j = i
      · subst j
        simp [singleMask, hi]
      · have hj : s j = false := by
          cases hv : s j
          · rfl
          · exact False.elim (htwo ⟨j, Ne.symm hji, hv⟩)
        simp [singleMask, hji, hj]
    subst s
    have ht := inputReject_transfer label σ F i
    have hf := hfar i
    unfold testerReject
    linarith

omit [Fintype A] [DecidableEq A] in
theorem selfCorrect_dictator (a : A) (x r : Cube A) :
    selfCorrect (fun f => f a) x r = x a := by
  change (r a ^^ (r a ^^ x a)) = x a
  cases r a <;> cases x a <;> rfl

omit [Nonempty K] in
theorem tester_perfect_completeness (label : A → K → Bool) (a : A) :
    testerReject label (label a) (fun f => f a) = 0 := by
  have hb : blrReject (fun f : Cube A => f a) = 0 := by
    simp [blrReject, cubeXor, disagree]
  have ha : andReject (fun f : Cube A => f a) = 0 := by
    simp [andReject, selfCorrect_dictator, cubeAnd, disagree]
  have hu : unitReject (fun f : Cube A => f a) = 0 := by
    simp [unitReject, selfCorrect_dictator, disagree]
  have hi : inputReject label (label a) (fun f => f a) = 0 := by
    simp [inputReject, selfCorrect_dictator, disagree]
  simp [testerReject, hb, ha, hu, hi]

def eventQueries (label : A → K → Bool) (kind : Fin 4) (k : K)
    (f g r₀ r₁ r₂ : Cube A) (slot : Fin 6) : K ⊕ Cube A :=
  if kind = 0 then
    if slot = 0 then .inr f else if slot = 1 then .inr g else .inr (cubeXor f g)
  else if kind = 1 then
    if slot = 0 then .inr r₀ else
    if slot = 1 then .inr (cubeXor r₀ (cubeAnd f g)) else
    if slot = 2 then .inr r₁ else
    if slot = 3 then .inr (cubeXor r₁ f) else
    if slot = 4 then .inr r₂ else .inr (cubeXor r₂ g)
  else if kind = 2 then
    if slot = 0 then .inr r₀ else .inr (cubeXor r₀ (fun _ => true))
  else
    if slot = 0 then .inl k else
    if slot = 1 then .inr r₀ else .inr (cubeXor r₀ (fun a => label a k))

def eventAccepts (kind : Fin 4) (b : Fin 6 → Bool) : Bool :=
  if kind = 0 then decide ((b 0 ^^ b 1) = b 2)
  else if kind = 1 then decide ((b 0 ^^ b 1) = ((b 2 ^^ b 3) && (b 4 ^^ b 5)))
  else if kind = 2 then (b 0 ^^ b 1)
  else decide (b 0 = (b 1 ^^ b 2))

def oracleAnswer (σ : K → Bool) (F : Cube A → Bool) : K ⊕ Cube A → Bool :=
  Sum.elim σ F

def eventReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (kind : Fin 4) (k : K) (f g r₀ r₁ r₂ : Cube A) : ℝ :=
  truth (!(eventAccepts kind (fun slot =>
    oracleAnswer σ F (eventQueries label kind k f g r₀ r₁ r₂ slot))))

theorem truth_not_decide_eq (a b : Bool) : truth (!(decide (a = b))) = disagree a b := by
  cases a <;> cases b <;> norm_num [truth, disagree]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_blr (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 0 k f g r₀ r₁ r₂ =
      disagree (F f ^^ F g) (F (cubeXor f g)) := by
  simp [eventReject, eventAccepts, eventQueries, oracleAnswer, truth_not_decide_eq]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_and (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 1 k f g r₀ r₁ r₂ =
      disagree (selfCorrect F (cubeAnd f g) r₀)
        (selfCorrect F f r₁ && selfCorrect F g r₂) := by
  simp [eventReject, eventAccepts, eventQueries, oracleAnswer,
    truth_not_decide_eq, selfCorrect]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_unit (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 2 k f g r₀ r₁ r₂ =
      disagree (selfCorrect F (fun _ => true) r₀) true := by
  simp only [eventReject, eventAccepts, eventQueries, oracleAnswer]
  norm_num
  change truth (!(selfCorrect F (fun _ => true) r₀)) =
    disagree (selfCorrect F (fun _ => true) r₀) true
  cases selfCorrect F (fun _ => true) r₀ <;> norm_num [truth, disagree]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem eventReject_input (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool)
    (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventReject label σ F 3 k f g r₀ r₁ r₂ =
      disagree (σ k) (selfCorrect F (fun a => label a k) r₀) := by
  simp [eventReject, eventAccepts, eventQueries, oracleAnswer,
    truth_not_decide_eq, selfCorrect]

omit [Fintype A] [DecidableEq A] [Fintype K] [Nonempty K] in
theorem event_perfect_completeness (label : A → K → Bool) (a : A)
    (kind : Fin 4) (k : K) (f g r₀ r₁ r₂ : Cube A) :
    eventAccepts kind (fun slot => oracleAnswer (label a) (fun x => x a)
      (eventQueries label kind k f g r₀ r₁ r₂ slot)) = true := by
  have hkind : kind = 0 ∨ kind = 1 ∨ kind = 2 ∨ kind = 3 := by omega
  have hreject : eventReject label (label a) (fun x => x a) kind k f g r₀ r₁ r₂ = 0 := by
    rcases hkind with h | h | h | h <;> subst kind
    · simp [eventReject_blr, cubeXor, disagree]
    · simp [eventReject_and, selfCorrect_dictator, cubeAnd, disagree]
    · simp [eventReject_unit, selfCorrect_dictator, disagree]
    · simp [eventReject_input, selfCorrect_dictator, disagree]
  unfold eventReject at hreject
  cases h : eventAccepts kind (fun slot => oracleAnswer (label a) (fun x => x a)
    (eventQueries label kind k f g r₀ r₁ r₂ slot)) <;> simp_all [truth]

def explicitTesterReject (label : A → K → Bool) (σ : K → Bool) (F : Cube A → Bool) : ℝ :=
  𝔼 kind : Fin 4, 𝔼 k, 𝔼 f, 𝔼 g, 𝔼 r₀, 𝔼 r₁, 𝔼 r₂,
    eventReject label σ F kind k f g r₀ r₁ r₂

theorem explicitTesterReject_eq (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) : explicitTesterReject label σ F = testerReject label σ F := by
  unfold explicitTesterReject
  rw [Fintype.expect_eq_sum_div_card]
  simp [Fin.sum_univ_succ, eventReject_blr, eventReject_and, eventReject_unit,
    eventReject_input, testerReject, blrReject, andReject, unitReject, inputReject, add_assoc]

theorem explicit_proximity_soundness (label : A → K → Bool) (σ : K → Bool)
    (F : Cube A → Bool) {δ : ℝ} (hδ : 0 < δ) (hδ' : δ ≤ 1 / 4)
    (hfar : ∀ a, δ ≤ distance σ (label a)) :
    δ / 256 ≤ explicitTesterReject label σ F := by
  rw [explicitTesterReject_eq]
  exact tester_proximity_soundness label σ F hδ hδ' hfar

end BinPackingGames.Foundations.PCP.AssignmentTester

namespace BinPackingGames.Foundations.PCP.AlphabetReduction

open BinPackingGames.Foundations.Hastad
open CodeComposition

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

abbrev LegalPair (accepts : A → A → Bool) :=
  {p : A × A // accepts p.1 p.2 = true}

abbrev InputCoordinate (A : Type*) := Bool × Cube A

def pairEncoding (accepts : A → A → Bool) (p : LegalPair accepts) :
    InputCoordinate A → Bool :=
  pairWord (codeword p.1.1) (codeword p.1.2)

def edgeTesterReject (accepts : A → A → Bool) (left right : Cube A → Bool)
    (F : Cube (LegalPair accepts) → Bool) : ℝ :=
  AssignmentTester.explicitTesterReject (pairEncoding accepts) (pairWord left right) F

theorem rejected_edge_tester_bound (accepts : A → A → Bool)
    (left right : Cube A → Bool)
    (F : Cube (LegalPair accepts) → Bool)
    (hbad : accepts (nearest left) (nearest right) = false) :
    1 / 2048 ≤ edgeTesterReject accepts left right F := by
  have hfar (p : LegalPair accepts) :
      (1 / 8 : ℝ) ≤ AssignmentTester.distance (pairWord left right) (pairEncoding accepts p) := by
    have h := rejected_pair_far accepts left right hbad p.1.1 p.1.2 p.2
    convert h using 1
    rfl
  have h := AssignmentTester.explicit_proximity_soundness (pairEncoding accepts)
    (pairWord left right) F (δ := 1 / 8) (by norm_num) (by norm_num) hfar
  norm_num at h
  exact h

omit [Nonempty A] in
theorem honest_edge_tester (accepts : A → A → Bool) (a b : A)
    (hab : accepts a b = true) :
    edgeTesterReject accepts (codeword a) (codeword b)
      (fun f => f ⟨(a, b), hab⟩) = 0 := by
  unfold edgeTesterReject
  rw [AssignmentTester.explicitTesterReject_eq]
  exact AssignmentTester.tester_perfect_completeness
    (pairEncoding accepts) ⟨(a, b), hab⟩

end BinPackingGames.Foundations.PCP.AlphabetReduction

namespace BinPackingGames.Foundations.PCP.AlphabetGraph

open BinPackingGames.Foundations.Hastad
open scoped BigOperators

abbrev Address (V E A : Type*) := (V × Cube A) ⊕ (E × Cube (A × A))

abbrev LocalEvent (A : Type*) := Fin 4 × AlphabetReduction.InputCoordinate A ×
  Cube (A × A) × Cube (A × A) × Cube (A × A) × Cube (A × A) × Cube (A × A)

abbrev Event (E A : Type*) := E × LocalEvent A

variable {V E A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]

abbrev legal (G : ConstraintGraph V E A) (e : E) (p : A × A) : Prop :=
  G.accepts e p.1 p.2 = true

instance legalDecidable (G : ConstraintGraph V E A) (e : E) :
    DecidablePred (legal G e) :=
  fun p => inferInstanceAs (Decidable (G.accepts e p.1 p.2 = true))

def restrictTape (G : ConstraintGraph V E A) (e : E) (f : Cube (A × A)) :
    Cube (AlphabetReduction.LegalPair (G.accepts e)) := UniformRestriction.restrict (legal G e) f

def extendTape (G : ConstraintGraph V E A) (e : E)
    (f : Cube (AlphabetReduction.LegalPair (G.accepts e))) : Cube (A × A) :=
  UniformRestriction.extend (legal G e) f

def globalizeQuery (G : ConstraintGraph V E A) (e : E) :
    AlphabetReduction.InputCoordinate A ⊕ Cube (AlphabetReduction.LegalPair (G.accepts e)) → Address V E A
  | .inl (side, tape) => .inl ((if side then G.head e else G.tail e), tape)
  | .inr tape => .inr (e, extendTape G e tape)

def verifier (G : ConstraintGraph V E A) : QueryIncidence.Verifier (Event E A) (Address V E A) 6 where
  query := fun (e, kind, k, f, g, r₀, r₁, r₂) slot =>
    globalizeQuery G e (AssignmentTester.eventQueries (AlphabetReduction.pairEncoding (G.accepts e)) kind k
      (restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
      (restrictTape G e r₁) (restrictTape G e r₂) slot)
  accepts := fun (_, kind, _, _, _, _, _, _) b => AssignmentTester.eventAccepts kind b

def vertexBlock (assignment : Address V E A → Bool) (v : V) : Cube A → Bool :=
  fun tape => assignment (.inl (v, tape))

def edgeOracle (G : ConstraintGraph V E A) (assignment : Address V E A → Bool) (e : E) :
    Cube (AlphabetReduction.LegalPair (G.accepts e)) → Bool :=
  fun tape => assignment (.inr (e, extendTape G e tape))

omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem oracleAnswer_globalize (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) (e : E)
    (x : AlphabetReduction.InputCoordinate A ⊕ Cube (AlphabetReduction.LegalPair (G.accepts e))) :
    assignment (globalizeQuery G e x) =
      AssignmentTester.oracleAnswer
        (CodeComposition.pairWord (vertexBlock assignment (G.tail e))
          (vertexBlock assignment (G.head e))) (edgeOracle G assignment e) x := by
  rcases x with ⟨side, tape⟩ | tape
  · cases side <;> rfl
  · rfl

omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem response_eq_local (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) (e : E) (kind : Fin 4)
    (k : AlphabetReduction.InputCoordinate A) (f g r₀ r₁ r₂ : Cube (A × A)) :
    QueryIncidence.response (verifier G) assignment (e, kind, k, f, g, r₀, r₁, r₂) =
      fun slot => AssignmentTester.oracleAnswer
        (CodeComposition.pairWord (vertexBlock assignment (G.tail e))
          (vertexBlock assignment (G.head e))) (edgeOracle G assignment e)
        (AssignmentTester.eventQueries (AlphabetReduction.pairEncoding (G.accepts e)) kind k
          (restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
          (restrictTape G e r₁) (restrictTape G e r₂) slot) := by
  funext slot
  exact oracleAnswer_globalize G assignment e _

def eventReject (G : ConstraintGraph V E A) (assignment : Address V E A → Bool)
    (event : Event E A) : ℝ :=
  AssignmentTester.truth (!((verifier G).accepts event (QueryIncidence.response (verifier G) assignment event)))

omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem eventReject_eq_local (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) (e : E) (kind : Fin 4)
    (k : AlphabetReduction.InputCoordinate A) (f g r₀ r₁ r₂ : Cube (A × A)) :
    eventReject G assignment (e, kind, k, f, g, r₀, r₁, r₂) =
      AssignmentTester.eventReject (AlphabetReduction.pairEncoding (G.accepts e))
        (CodeComposition.pairWord (vertexBlock assignment (G.tail e))
          (vertexBlock assignment (G.head e))) (edgeOracle G assignment e) kind k
        (restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
        (restrictTape G e r₁) (restrictTape G e r₂) := by
  unfold eventReject
  rw [response_eq_local]
  rfl

theorem expect_prod {I J : Type*} [Fintype I] [Fintype J] (f : I × J → ℝ) :
    (𝔼 z : I × J, f z) = 𝔼 i : I, 𝔼 j : J, f (i, j) := by
  simpa only [Finset.univ_product_univ] using
    Finset.expect_product (Finset.univ : Finset I) (Finset.univ : Finset J) f

def localAverage (G : ConstraintGraph V E A) (assignment : Address V E A → Bool)
    (e : E) : ℝ := 𝔼 sample : LocalEvent A, eventReject G assignment (e, sample)

theorem expect_restrict_five {B : Type*} [Fintype B] [DecidableEq B]
    (p : B → Prop) [DecidablePred p]
    (h : ({b // p b} → Bool) → ({b // p b} → Bool) → ({b // p b} → Bool) →
      ({b // p b} → Bool) → ({b // p b} → Bool) → ℝ) :
    (𝔼 f : B → Bool, 𝔼 g : B → Bool, 𝔼 r₀ : B → Bool,
      𝔼 r₁ : B → Bool, 𝔼 r₂ : B → Bool,
        h (UniformRestriction.restrict p f) (UniformRestriction.restrict p g)
          (UniformRestriction.restrict p r₀) (UniformRestriction.restrict p r₁)
          (UniformRestriction.restrict p r₂)) =
    𝔼 f : {b // p b} → Bool, 𝔼 g : {b // p b} → Bool,
      𝔼 r₀ : {b // p b} → Bool, 𝔼 r₁ : {b // p b} → Bool,
        𝔼 r₂ : {b // p b} → Bool, h f g r₀ r₁ r₂ := by
  rw [UniformRestriction.expect_restrict p
    (fun f => 𝔼 g : B → Bool, 𝔼 r₀ : B → Bool, 𝔼 r₁ : B → Bool,
      𝔼 r₂ : B → Bool, h f (UniformRestriction.restrict p g)
        (UniformRestriction.restrict p r₀) (UniformRestriction.restrict p r₁)
        (UniformRestriction.restrict p r₂))]
  apply Finset.expect_congr rfl
  intro f _
  rw [UniformRestriction.expect_restrict p
    (fun g => 𝔼 r₀ : B → Bool, 𝔼 r₁ : B → Bool, 𝔼 r₂ : B → Bool,
      h f g (UniformRestriction.restrict p r₀) (UniformRestriction.restrict p r₁)
        (UniformRestriction.restrict p r₂))]
  apply Finset.expect_congr rfl
  intro g _
  rw [UniformRestriction.expect_restrict p
    (fun r₀ => 𝔼 r₁ : B → Bool, 𝔼 r₂ : B → Bool,
      h f g r₀ (UniformRestriction.restrict p r₁) (UniformRestriction.restrict p r₂))]
  apply Finset.expect_congr rfl
  intro r₀ _
  rw [UniformRestriction.expect_restrict p
    (fun r₁ => 𝔼 r₂ : B → Bool, h f g r₀ r₁ (UniformRestriction.restrict p r₂))]
  apply Finset.expect_congr rfl
  intro r₁ _
  exact UniformRestriction.expect_restrict p (h f g r₀ r₁)

omit [Nonempty A] in
theorem localAverage_eq_tester (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) (e : E) :
    localAverage G assignment e = AlphabetReduction.edgeTesterReject (G.accepts e)
      (vertexBlock assignment (G.tail e)) (vertexBlock assignment (G.head e))
      (edgeOracle G assignment e) := by
  unfold localAverage AlphabetReduction.edgeTesterReject AssignmentTester.explicitTesterReject
  simp only [LocalEvent, expect_prod, eventReject_eq_local, restrictTape]
  apply Finset.expect_congr rfl
  intro kind _
  apply Finset.expect_congr rfl
  intro side _
  apply Finset.expect_congr rfl
  intro tape _
  exact expect_restrict_five (legal G e)
    (AssignmentTester.eventReject (AlphabetReduction.pairEncoding (G.accepts e))
      (CodeComposition.pairWord (vertexBlock assignment (G.tail e))
        (vertexBlock assignment (G.head e))) (edgeOracle G assignment e) kind (side, tape))

def average [Fintype E] (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) : ℝ :=
  𝔼 event : Event E A, eventReject G assignment event

omit [Nonempty A] in
theorem average_eq_testers [Fintype E] (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) :
    average G assignment = 𝔼 e : E, AlphabetReduction.edgeTesterReject (G.accepts e)
      (vertexBlock assignment (G.tail e)) (vertexBlock assignment (G.head e))
      (edgeOracle G assignment e) := by
  unfold average
  rw [expect_prod]
  exact Finset.expect_congr rfl (fun e _ => localAverage_eq_tester G assignment e)

def decodedLabeling (assignment : Address V E A → Bool) : V → A :=
  fun v => CodeComposition.nearest (vertexBlock assignment v)

theorem local_gap (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) (e : E) :
    AssignmentTester.truth (!(G.edgeSatisfied (decodedLabeling assignment) e)) / 2048 ≤
      localAverage G assignment e := by
  cases h : G.edgeSatisfied (decodedLabeling assignment) e with
  | false =>
      rw [localAverage_eq_tester]
      simpa [AssignmentTester.truth] using
        AlphabetReduction.rejected_edge_tester_bound (G.accepts e)
          (vertexBlock assignment (G.tail e)) (vertexBlock assignment (G.head e))
          (edgeOracle G assignment e) h
  | true =>
      simp only [Bool.not_true, AssignmentTester.truth, Bool.false_eq_true, ite_false, zero_div]
      apply Finset.expect_nonneg
      intro sample _
      exact AssignmentTester.truth_nonneg _

theorem average_gap [Fintype E] (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) :
    (𝔼 e : E, AssignmentTester.truth (!(G.edgeSatisfied (decodedLabeling assignment) e))) / 2048 ≤
      average G assignment := by
  unfold average
  rw [expect_prod, Finset.expect_div]
  exact Finset.expect_le_expect (fun e _ => local_gap G assignment e)

theorem expect_rejection_eq_count {I : Type*} [Fintype I] (b : I → Bool) :
    (𝔼 i : I, AssignmentTester.truth (!(b i))) =
      ((Finset.univ.filter (fun i => b i = false)).card : ℝ) / Fintype.card I := by
  rw [Fintype.expect_eq_sum_div_card]
  congr 1
  calc
    (∑ i : I, AssignmentTester.truth (!(b i))) = ∑ i : I, if b i = false then (1 : ℝ) else 0 := by
      apply Finset.sum_congr rfl
      intro i _
      cases b i <;> norm_num [AssignmentTester.truth]
    _ = _ := Finset.sum_boole _ _

omit [Nonempty A] in
theorem average_eq_count [Fintype E] (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) :
    average G assignment = (QueryIncidence.verifierRejectionCount (verifier G) assignment : ℝ) /
      Fintype.card (Event E A) :=
  expect_rejection_eq_count _

omit [Nonempty A] in
@[simp] theorem card_event [Fintype E] :
    Fintype.card (Event E A) = Fintype.card E * Fintype.card (LocalEvent A) :=
  Fintype.card_prod _ _

omit [Nonempty A] in
theorem card_localEvent :
    Fintype.card (LocalEvent A) =
      4 * (2 * 2 ^ Fintype.card A) * (2 ^ (Fintype.card A * Fintype.card A)) ^ 5 := by
  simp [LocalEvent, AlphabetReduction.InputCoordinate, Cube, Fintype.card_prod,
    pow_succ, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]

omit [Nonempty A] in
theorem card_address [Fintype V] [Fintype E] :
    Fintype.card (Address V E A) =
      Fintype.card V * 2 ^ Fintype.card A +
        Fintype.card E * 2 ^ (Fintype.card A * Fintype.card A) := by
  simp [Address, Cube, Fintype.card_sum, Fintype.card_prod]

theorem rejection_count_bound [Fintype E] (G : ConstraintGraph V E A)
    (assignment : Address V E A → Bool) :
    Fintype.card (LocalEvent A) * G.rejectionCount (decodedLabeling assignment) ≤
      2048 * QueryIncidence.verifierRejectionCount (verifier G) assignment := by
  classical
  cases isEmpty_or_nonempty E with
  | inl empty =>
      let := empty
      simp [ConstraintGraph.rejectionCount, ConstraintGraph.rejectedDarts]
  | inr inhabited =>
      let := inhabited
      have h := average_gap G assignment
      rw [average_eq_count, expect_rejection_eq_count, card_event, Nat.cast_mul,
        div_div] at h
      have he : (0 : ℝ) < Fintype.card E := by exact_mod_cast Fintype.card_pos
      have hl : (0 : ℝ) < Fintype.card (LocalEvent A) := by
        exact_mod_cast Fintype.card_pos
      have hc := (div_le_div_iff₀ (mul_pos he (by norm_num)) (mul_pos he hl)).mp h
      have hc' :
          ((Fintype.card (LocalEvent A) : ℝ) *
            (G.rejectionCount (decodedLabeling assignment) : ℝ)) * Fintype.card E ≤
          (2048 * (QueryIncidence.verifierRejectionCount (verifier G) assignment : ℝ)) * Fintype.card E := by
        simpa only [ConstraintGraph.rejectionCount, ConstraintGraph.rejectedDarts,
          mul_assoc, mul_comm, mul_left_comm] using hc
      have result := (mul_le_mul_iff_left₀ he).mp hc'
      exact_mod_cast result

def honestAssignment (G : ConstraintGraph V E A) (labeling : V → A) : Address V E A → Bool
  | .inl (v, tape) => tape (labeling v)
  | .inr (e, tape) => tape (labeling (G.tail e), labeling (G.head e))

omit [Fintype A] [DecidableEq A] [Nonempty A] in
@[simp] theorem honest_vertexBlock (G : ConstraintGraph V E A) (labeling : V → A) (v : V) :
    vertexBlock (honestAssignment G labeling) v = CodeComposition.codeword (labeling v) := rfl

omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem honest_edgeOracle (G : ConstraintGraph V E A) (labeling : V → A) (e : E)
    (h : G.edgeSatisfied labeling e = true) :
    edgeOracle G (honestAssignment G labeling) e =
      fun tape => tape ⟨(labeling (G.tail e), labeling (G.head e)), h⟩ := by
  funext tape
  exact UniformRestriction.extend_apply_of_mem (legal G e) tape
    (labeling (G.tail e), labeling (G.head e)) h

omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem honest_event (G : ConstraintGraph V E A) (labeling : V → A)
    (satisfied : ∀ e, G.edgeSatisfied labeling e = true) (event : Event E A) :
    (verifier G).accepts event (QueryIncidence.response (verifier G) (honestAssignment G labeling) event) =
      true := by
  rcases event with ⟨e, kind, k, f, g, r₀, r₁, r₂⟩
  rw [response_eq_local, honest_edgeOracle G labeling e (satisfied e)]
  simp only [honest_vertexBlock]
  exact AssignmentTester.event_perfect_completeness (AlphabetReduction.pairEncoding (G.accepts e))
    ⟨(labeling (G.tail e), labeling (G.head e)), satisfied e⟩ kind k
    (restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
    (restrictTape G e r₁) (restrictTape G e r₂)

def graph [DecidableEq V] [DecidableEq E] (G : ConstraintGraph V E A) :
    ConstraintGraph (QueryIncidence.Vertex (Event E A) (Address V E A))
      (QueryIncidence.Dart (Event E A) 6) (QueryIncidence.Label 6) := QueryIncidence.graph (verifier G) (by omega)

omit [DecidableEq A] [Nonempty A] in
theorem perfect_completeness [DecidableEq V] [DecidableEq E] (G : ConstraintGraph V E A)
    (satisfied : G.Satisfiable) : (graph G).Satisfiable := by
  obtain ⟨labeling, h⟩ := satisfied
  exact QueryIncidence.satisfiable_of_verifier_satisfiable (verifier G) (by omega)
    ⟨honestAssignment G labeling, honest_event G labeling h⟩

theorem gap_transfer [Fintype E] [DecidableEq V] [DecidableEq E]
    (G : ConstraintGraph V E A) (a b : Nat)
    (source : ∀ labeling : V → A,
      a * Fintype.card E ≤ b * G.rejectionCount labeling)
    (labeling : QueryIncidence.Vertex (Event E A) (Address V E A) → QueryIncidence.Label 6) :
    a * Fintype.card (QueryIncidence.Dart (Event E A) 6) ≤
      (b * 12288) * (graph G).rejectionCount labeling := by
  have verifierGap (assignment : Address V E A → Bool) :
      a * Fintype.card (Event E A) ≤
        (b * 2048) * QueryIncidence.verifierRejectionCount (verifier G) assignment := by
    rw [card_event]
    calc
      a * (Fintype.card E * Fintype.card (LocalEvent A)) =
          Fintype.card (LocalEvent A) * (a * Fintype.card E) := by ac_rfl
      _ ≤ Fintype.card (LocalEvent A) * (b * G.rejectionCount (decodedLabeling assignment)) :=
        Nat.mul_le_mul_left _ (source (decodedLabeling assignment))
      _ = b * (Fintype.card (LocalEvent A) * G.rejectionCount (decodedLabeling assignment)) := by
        ac_rfl
      _ ≤ b * (2048 * QueryIncidence.verifierRejectionCount (verifier G) assignment) :=
        Nat.mul_le_mul_left _ (rejection_count_bound G assignment)
      _ = (b * 2048) * QueryIncidence.verifierRejectionCount (verifier G) assignment := by ac_rfl
  have h := QueryIncidence.gap_transfer (verifier G) (by omega) a (b * 2048) verifierGap labeling
  have hcoef : b * 2048 * 6 = b * 12288 := by omega
  rw [hcoef] at h
  exact h

theorem alphabet_card : Fintype.card (QueryIncidence.Label 6) = 64 := QueryIncidence.six_query_alphabet

end BinPackingGames.Foundations.PCP.AlphabetGraph

namespace BinPackingGames.Foundations.PCP.AlphabetTable.Queries

open BinPackingGames.Foundations.Hastad

abbrev RawQuery (q : Nat) :=
  (Bool × Cube (Fin q)) ⊕ Cube (Fin q × Fin q)

abbrev LocalEvent (q : Nat) := AlphabetGraph.LocalEvent (Fin q)

def mask {q : Nat} (P : Fin q → Fin q → Bool) (f : Cube (Fin q × Fin q)) :
    Cube (Fin q × Fin q) := fun p => if P p.1 p.2 then f p else false

def xorTape {A : Type*} (f g : Cube A) : Cube A := fun a => f a ^^ g a

def andTape {A : Type*} (f g : Cube A) : Cube A := fun a => f a && g a

def inputPairTape {q : Nat} (k : Bool × Cube (Fin q)) : Cube (Fin q × Fin q) :=
  fun p => if k.1 then k.2 p.2 else k.2 p.1

def query {q : Nat} (P : Fin q → Fin q → Bool) (event : LocalEvent q)
    (slot : Fin 6) : RawQuery q :=
  match event with
  | (kind, k, f, g, r₀, r₁, r₂) =>
    if kind = 0 then
      if slot = 0 then .inr (mask P f) else
      if slot = 1 then .inr (mask P g) else .inr (mask P (xorTape f g))
    else if kind = 1 then
      if slot = 0 then .inr (mask P r₀) else
      if slot = 1 then .inr (mask P (xorTape r₀ (andTape f g))) else
      if slot = 2 then .inr (mask P r₁) else
      if slot = 3 then .inr (mask P (xorTape r₁ f)) else
      if slot = 4 then .inr (mask P r₂) else .inr (mask P (xorTape r₂ g))
    else if kind = 2 then
      if slot = 0 then .inr (mask P r₀) else
        .inr (mask P (xorTape r₀ (fun _ => true)))
    else
      if slot = 0 then .inl k else
      if slot = 1 then .inr (mask P r₀) else
        .inr (mask P (xorTape r₀ (inputPairTape k)))

def globalize {q : Nat} {V E : Type*} (tail head : V) (edge : E) :
    RawQuery q → AlphabetGraph.Address V E (Fin q)
  | .inl (side, tape) => .inl ((if side then head else tail), tape)
  | .inr tape => .inr (edge, tape)

@[simp] theorem mask_idempotent {q : Nat} (P : Fin q → Fin q → Bool)
    (f : Cube (Fin q × Fin q)) : mask P (mask P f) = mask P f := by
  funext p
  cases hp : P p.1 p.2 <;> simp [mask, hp]

theorem mask_of_empty {q : Nat} (P : Fin q → Fin q → Bool)
    (empty : ∀ a b, P a b = false) (f : Cube (Fin q × Fin q)) :
    mask P f = fun _ => false := by
  funext p
  simp [mask, empty]

theorem query_is_masked {q : Nat} (P : Fin q → Fin q → Bool)
    (event : LocalEvent q) (slot : Fin 6) :
    match query P event slot with
    | .inl _ => True
    | .inr tape => mask P tape = tape := by
  rcases event with ⟨kind, k, f, g, r₀, r₁, r₂⟩
  by_cases hk₀ : kind = 0 <;> by_cases hk₁ : kind = 1 <;> by_cases hk₂ : kind = 2 <;>
    simp only [query, hk₀, hk₁, hk₂, ite_true, ite_false] <;>
    split_ifs <;> simp

def extendQuery {q : Nat} (P : Fin q → Fin q → Bool) :
    AlphabetReduction.InputCoordinate (Fin q) ⊕ Cube (AlphabetReduction.LegalPair P) →
      RawQuery q
  | .inl k => .inl k
  | .inr tape => .inr (UniformRestriction.extend (fun p : Fin q × Fin q => P p.1 p.2 = true) tape)

theorem query_eq_extend_eventQueries {q : Nat} (P : Fin q → Fin q → Bool)
    (kind : Fin 4) (k : AlphabetReduction.InputCoordinate (Fin q))
    (f g r₀ r₁ r₂ : Cube (Fin q × Fin q)) (slot : Fin 6) :
    query P (kind, k, f, g, r₀, r₁, r₂) slot =
      extendQuery P (AssignmentTester.eventQueries (AlphabetReduction.pairEncoding P) kind k
        (UniformRestriction.restrict (fun p : Fin q × Fin q => P p.1 p.2 = true) f)
        (UniformRestriction.restrict (fun p : Fin q × Fin q => P p.1 p.2 = true) g)
        (UniformRestriction.restrict (fun p : Fin q × Fin q => P p.1 p.2 = true) r₀)
        (UniformRestriction.restrict (fun p : Fin q × Fin q => P p.1 p.2 = true) r₁)
        (UniformRestriction.restrict (fun p : Fin q × Fin q => P p.1 p.2 = true) r₂) slot) := by
  by_cases hk₀ : kind = 0 <;> by_cases hk₁ : kind = 1 <;> by_cases hk₂ : kind = 2 <;>
    simp only [query, AssignmentTester.eventQueries, hk₀, hk₁, hk₂, ite_true, ite_false]
  all_goals split_ifs <;> simp_all only [extendQuery]
  all_goals rfl

theorem globalize_extendQuery {q : Nat} {V E : Type*}
    (G : ConstraintGraph V E (Fin q)) (e : E)
    (x : AlphabetReduction.InputCoordinate (Fin q) ⊕
      Cube (AlphabetReduction.LegalPair (G.accepts e))) :
    globalize (G.tail e) (G.head e) e (extendQuery (G.accepts e) x) =
      AlphabetGraph.globalizeQuery G e x := by
  cases x with
  | inl k => cases k; rfl
  | inr tape => rfl

theorem globalize_query {q : Nat} {V E : Type*}
    (G : ConstraintGraph V E (Fin q)) (e : E) (event : LocalEvent q) (slot : Fin 6) :
    globalize (G.tail e) (G.head e) e (query (G.accepts e) event slot) =
      (AlphabetGraph.verifier G).query (e, event) slot := by
  rcases event with ⟨kind, k, f, g, r₀, r₁, r₂⟩
  rw [query_eq_extend_eventQueries]
  exact globalize_extendQuery G e _

end BinPackingGames.Foundations.PCP.AlphabetTable.Queries

end

end OAI