Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.MaxCut.PCP.Variation

namespace OAI

/-!
The finite calculation behind v2 Lemma `projected-tv`. The three singleton
lines are averaged before the divergence is taken; projection choices are
not part of the compared sample space. Coordinates are independently sampled
occurrences, irrespective of whether their equation IDs coincide.
-/

namespace MaxCutGames.Decoder.SparseLaw

open scoped BigOperators
open MaxCutGames.Foundations.Information

noncomputable section

variable {α : Type*} [Fintype α]

def uniformWeights (α : Type*) [Fintype α] : α → ℝ :=
  fun _ => 1 / Fintype.card α

theorem uniformWeights_pos [Nonempty α] (a : α) : 0 < uniformWeights α a := by
  exact div_pos (by norm_num) (Nat.cast_pos.mpr Fintype.card_pos)

theorem uniformWeights_isProbability [Nonempty α] :
    IsProbability (uniformWeights α) := by
  constructor
  · exact fun a => (uniformWeights_pos a).le
  · simp [uniformWeights, Fintype.card_ne_zero]

/-- Pearson divergence against the uniform law, written with its constant
denominator cleared. `uniformChiSquare_eq_pearson` identifies the actual
likelihood-ratio definition. -/
def uniformChiSquare (p : α → ℝ) : ℝ :=
  (Fintype.card α : ℝ) * ∑ a, (p a - uniformWeights α a) ^ 2

theorem uniformChiSquare_eq_pearson [Nonempty α] (p : α → ℝ) :
    uniformChiSquare p =
      ∑ a, uniformWeights α a * (p a / uniformWeights α a - 1) ^ 2 := by
  rw [uniformChiSquare, Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro a _
  have hn : (Fintype.card α : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
  simp only [uniformWeights]
  field_simp [hn]

theorem uniformChiSquare_eq_secondMoment [Nonempty α] (p : α → ℝ)
    (hp : IsProbability p) :
    uniformChiSquare p = (Fintype.card α : ℝ) * (∑ a, p a ^ 2) - 1 := by
  have hn : (Fintype.card α : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
  have hpoint (a : α) :
      (Fintype.card α : ℝ) * (p a - uniformWeights α a) ^ 2 =
        (Fintype.card α : ℝ) * p a ^ 2 - 2 * p a + uniformWeights α a := by
    simp only [uniformWeights]
    field_simp [hn]
    ring
  rw [uniformChiSquare, Finset.mul_sum]
  simp only [hpoint, Finset.sum_add_distrib,
    Finset.sum_sub_distrib, ← Finset.mul_sum, hp.2,
    (uniformWeights_isProbability (α := α)).2]
  ring

theorem uniformChiSquare_nonneg (p : α → ℝ) : 0 ≤ uniformChiSquare p := by
  unfold uniformChiSquare
  positivity

def mixture (β : ℝ) (p : α → ℝ) : α → ℝ :=
  fun a => (1 - β) * uniformWeights α a + β * p a

theorem mixture_isProbability [Nonempty α] (p : α → ℝ) (hp : IsProbability p)
    {β : ℝ} (hβ : 0 ≤ β) (hβ' : β ≤ 1) : IsProbability (mixture β p) := by
  constructor
  · intro a
    exact add_nonneg (mul_nonneg (sub_nonneg.mpr hβ') (uniformWeights_pos a).le)
      (mul_nonneg hβ (hp.1 a))
  · simp only [mixture, Finset.sum_add_distrib, ← Finset.mul_sum, hp.2,
      (uniformWeights_isProbability (α := α)).2]
    ring

theorem uniformChiSquare_mixture (p : α → ℝ) (β : ℝ) :
    uniformChiSquare (mixture β p) = β ^ 2 * uniformChiSquare p := by
  have hpoint (a : α) :
      (mixture β p a - uniformWeights α a) ^ 2 =
        β ^ 2 * (p a - uniformWeights α a) ^ 2 := by
    unfold mixture
    ring
  simp only [uniformChiSquare, hpoint, ← Finset.mul_sum]
  ring

theorem totalVariation_le_half_sqrt (p : α → ℝ) :
    totalVariation p (uniformWeights α) ≤ Real.sqrt (uniformChiSquare p) / 2 := by
  have hcs := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset α)
    (fun _ => (1 : ℝ)) (fun a => |p a - uniformWeights α a|)
  simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
    mul_one, sq_abs] at hcs
  have hs := Real.le_sqrt_of_sq_le hcs
  exact div_le_div_of_nonneg_right hs (by norm_num : (0 : ℝ) ≤ 2)

def independentWeights (p : α → ℝ) (k : ℕ) : (Fin k → α) → ℝ :=
  fun x => ∏ i, p (x i)

theorem independentWeights_isProbability (p : α → ℝ) (hp : IsProbability p)
    (k : ℕ) : IsProbability (independentWeights p k) := by
  constructor
  · intro x
    exact Finset.prod_nonneg fun i _ => hp.1 (x i)
  · change (∑ x : Fin k → α, ∏ i, p (x i)) = 1
    rw [← Fintype.sum_pow, hp.2, one_pow]

theorem uniformChiSquare_independent [Nonempty α] (p : α → ℝ)
    (hp : IsProbability p) (k : ℕ) :
    uniformChiSquare (independentWeights p k) = (1 + uniformChiSquare p) ^ k - 1 := by
  rw [uniformChiSquare_eq_secondMoment _ (independentWeights_isProbability p hp k),
    uniformChiSquare_eq_secondMoment p hp]
  have hsecond : (∑ x : Fin k → α, independentWeights p k x ^ 2) =
      (∑ a, p a ^ 2) ^ k := by
    simp only [independentWeights, ← Finset.prod_pow]
    exact (Fintype.sum_pow (fun a => p a ^ 2) k).symm
  rw [hsecond]
  simp only [Fintype.card_fun, Fintype.card_fin, Nat.cast_pow]
  rw [← mul_pow]
  exact congrArg (fun t : ℝ => t ^ k - 1) (by ring)

/-- A finite bound sufficient for the cube asymptotic, without introducing a
limit theorem or choosing the alphabet after the repetition count. -/
theorem pow_mul_one_sub_le_one (x : ℝ) (hx : 0 ≤ x) (k : ℕ) :
    (1 + x) ^ k * (1 - k * x) ≤ 1 := by
  induction k with
  | zero => simp
  | succ k ih =>
    have hp : 0 ≤ (1 + x) ^ k := pow_nonneg (by linarith) _
    have herr : 0 ≤ (1 + x) ^ k * ((k : ℝ) + 1) * x ^ 2 :=
      mul_nonneg (mul_nonneg hp (by positivity)) (sq_nonneg x)
    rw [pow_succ, Nat.cast_add, Nat.cast_one]
    nlinarith

theorem pow_sub_one_le_fraction (x : ℝ) (hx : 0 ≤ x) (k : ℕ)
    (hk : (k : ℝ) * x < 1) :
    (1 + x) ^ k - 1 ≤ (k : ℝ) * x / (1 - (k : ℝ) * x) := by
  apply (le_div_iff₀ (sub_pos.mpr hk)).2
  have h := pow_mul_one_sub_le_one x hx k
  nlinarith

theorem pow_sub_one_le_twice (x : ℝ) (hx : 0 ≤ x) (k : ℕ)
    (hk : (k : ℝ) * x ≤ 1 / 2) :
    (1 + x) ^ k - 1 ≤ 2 * (k : ℝ) * x := by
  have hk' : (k : ℝ) * x < 1 := by linarith
  refine (pow_sub_one_le_fraction x hx k hk').trans ?_
  apply (div_le_iff₀ (sub_pos.mpr hk')).2
  have hn : 0 ≤ (k : ℝ) * x := mul_nonneg (Nat.cast_nonneg _) hx
  nlinarith

section Singleton

variable (V : Type*) [Fintype V] [Zero V] [DecidableEq V]

/-- At the origin all three singleton lines contribute. -/
def lineCount (z : V × V) : ℝ :=
  (if z.2 = 0 then 1 else 0) + (if z.1 = 0 then 1 else 0) +
    (if z.1 = z.2 then 1 else 0)

/-- The average of the laws of `(a,0)`, `(0,a)`, and `(a,a)` for uniform `a`. -/
def singletonPairWeights (z : V × V) : ℝ := lineCount V z / (3 * Fintype.card V)

/-- Explicit finite pushforward of the three equally likely singleton choices.
The same uniform slope supplies both entries of the diagonal choice. -/
def singletonSamplerWeights (z : V × V) : ℝ :=
  ((∑ a : V, if (a, 0) = z then uniformWeights V a else 0) +
    (∑ a : V, if (0, a) = z then uniformWeights V a else 0) +
    (∑ a : V, if (a, a) = z then uniformWeights V a else 0)) / 3

theorem singletonSamplerWeights_eq (z : V × V) :
    singletonSamplerWeights V z = singletonPairWeights V z := by
  classical
  rcases z with ⟨x, y⟩
  have hleft : (∑ a : V, if (a, 0) = (x, y) then uniformWeights V a else 0) =
      if y = 0 then 1 / (Fintype.card V : ℝ) else 0 := by
    by_cases hy : y = 0
    · simp [hy, uniformWeights]
    · simp [Prod.mk.injEq, Ne.symm hy, hy]
  have hright : (∑ a : V, if (0, a) = (x, y) then uniformWeights V a else 0) =
      if x = 0 then 1 / (Fintype.card V : ℝ) else 0 := by
    by_cases hx : x = 0
    · simp [hx, uniformWeights]
    · simp [Prod.mk.injEq, Ne.symm hx, hx]
  have hdiag : (∑ a : V, if (a, a) = (x, y) then uniformWeights V a else 0) =
      if x = y then 1 / (Fintype.card V : ℝ) else 0 := by
    by_cases hxy : x = y
    · simp [hxy, uniformWeights]
    · have hnone (a : V) : (a, a) ≠ (x, y) := by
        intro ha
        have he := Prod.mk.inj ha
        exact hxy (he.1.symm.trans he.2)
      simp [hnone, hxy]
  simp only [singletonSamplerWeights, hleft, hright, hdiag,
    singletonPairWeights, lineCount]
  split_ifs <;> ring

theorem lineCount_sum : ∑ z : V × V, lineCount V z = 3 * Fintype.card V := by
  have h₁ : (∑ z : V × V, if z.2 = 0 then (1 : ℝ) else 0) = Fintype.card V := by
    rw [Fintype.sum_prod_type]
    simp
  have h₂ : (∑ z : V × V, if z.1 = 0 then (1 : ℝ) else 0) = Fintype.card V := by
    rw [Fintype.sum_prod_type, Finset.sum_comm]
    simp
  have h₃ : (∑ z : V × V, if z.1 = z.2 then (1 : ℝ) else 0) = Fintype.card V := by
    rw [Fintype.sum_prod_type]
    simp
  simp only [lineCount, Finset.sum_add_distrib, h₁, h₂, h₃]
  ring

omit [Fintype V] in
theorem lineCount_sq (z : V × V) :
    lineCount V z ^ 2 = lineCount V z + (if z.1 = 0 ∧ z.2 = 0 then 6 else 0) := by
  by_cases hx : z.1 = 0 <;> by_cases hy : z.2 = 0 <;>
    by_cases hxy : z.1 = z.2 <;> simp_all [lineCount] ; norm_num

theorem lineCount_sq_sum :
    ∑ z : V × V, lineCount V z ^ 2 = 3 * Fintype.card V + 6 := by
  simp only [lineCount_sq, Finset.sum_add_distrib, lineCount_sum]
  congr 1
  rw [Fintype.sum_prod_type]
  rw [Finset.sum_eq_single (0 : V)]
  · simp
  · intro x _ hx
    simp [hx]
  · simp

theorem singletonPairWeights_isProbability : IsProbability (singletonPairWeights V) := by
  have hn : (0 : ℝ) < Fintype.card V := Nat.cast_pos.mpr Fintype.card_pos
  constructor
  · intro z
    unfold singletonPairWeights lineCount
    positivity
  · simp only [singletonPairWeights, div_eq_mul_inv, ← Finset.sum_mul, lineCount_sum]
    exact mul_inv_cancel₀ (ne_of_gt (mul_pos (by norm_num) hn))

theorem singletonPairWeights_secondMoment :
    ∑ z : V × V, singletonPairWeights V z ^ 2 =
      (3 * Fintype.card V + 6) / (3 * Fintype.card V) ^ 2 := by
  simp only [singletonPairWeights, div_pow]
  simp only [div_eq_mul_inv, ← Finset.sum_mul, lineCount_sq_sum]

theorem singletonPairWeights_chiSquare :
    uniformChiSquare (singletonPairWeights V) = ((Fintype.card V : ℝ) - 1) / 3 := by
  rw [uniformChiSquare_eq_secondMoment _ (singletonPairWeights_isProbability V),
    singletonPairWeights_secondMoment]
  simp only [Fintype.card_prod, Nat.cast_mul]
  have hn : (Fintype.card V : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
  field_simp [hn]
  ring

/-- Exact finite divergence of the padded slope law. -/
theorem paddedSlope_chiSquare (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) (k : ℕ) :
    uniformChiSquare (independentWeights (mixture β (singletonPairWeights V)) k) =
      (1 + β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ^ k - 1 := by
  rw [uniformChiSquare_independent _
    (mixture_isProbability _ (singletonPairWeights_isProbability V) hβ hβ'),
    uniformChiSquare_mixture, singletonPairWeights_chiSquare]
  exact congrArg (fun t : ℝ => (1 + t) ^ k - 1) (by ring)

theorem paddedSlope_totalVariation (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) (k : ℕ) :
    totalVariation (independentWeights (mixture β (singletonPairWeights V)) k)
      (uniformWeights (Fin k → V × V)) ≤
        Real.sqrt ((1 + β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ^ k - 1) / 2 := by
  simpa only [paddedSlope_chiSquare V β hβ hβ' k] using
    totalVariation_le_half_sqrt (independentWeights (mixture β (singletonPairWeights V)) k)

theorem paddedSlope_totalVariation_small (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1)
    (k : ℕ)
    (hk : (k : ℝ) * (β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ≤ 1 / 2) :
    totalVariation (independentWeights (mixture β (singletonPairWeights V)) k)
      (uniformWeights (Fin k → V × V)) ≤
        Real.sqrt (2 * (k : ℝ) * (β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3)) / 2 := by
  have hn : (1 : ℝ) ≤ Fintype.card V := by
    exact_mod_cast (Nat.succ_le_iff.mpr (Fintype.card_pos : 0 < Fintype.card V))
  refine (paddedSlope_totalVariation V β hβ hβ' k).trans ?_
  apply div_le_div_of_nonneg_right _ (by norm_num)
  apply Real.sqrt_le_sqrt
  exact pow_sub_one_le_twice _ (by positivity) _ hk

theorem cubeSlope_totalVariation (m : ℕ) (hm : 0 < m)
    (hsize : 2 * ((Fintype.card V : ℝ) - 1) / 3 ≤ m) :
    totalVariation
      (independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3))
      (uniformWeights (Fin (m ^ 3) → V × V)) ≤
        Real.sqrt (2 * ((Fintype.card V : ℝ) - 1) / (3 * m)) / 2 := by
  have hmpos : (0 : ℝ) < m := Nat.cast_pos.mpr hm
  have hmone : (1 : ℝ) ≤ m := by exact_mod_cast (Nat.succ_le_iff.mpr hm)
  have hbeta : (1 : ℝ) / (m : ℝ) ^ 2 ≤ 1 := by
    apply (div_le_iff₀ (sq_pos_of_pos hmpos)).2
    nlinarith
  have hproduct : ((m ^ 3 : ℕ) : ℝ) *
      ((1 / (m : ℝ) ^ 2) ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) =
        ((Fintype.card V : ℝ) - 1) / (3 * m) := by
    push_cast
    field_simp [hmpos.ne']

  have hsmall : ((m ^ 3 : ℕ) : ℝ) *
      ((1 / (m : ℝ) ^ 2) ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ≤ 1 / 2 := by
    rw [hproduct]
    apply (div_le_iff₀ (mul_pos (by norm_num) hmpos)).2
    linarith
  have h := paddedSlope_totalVariation_small V (1 / (m : ℝ) ^ 2)
    (by positivity) hbeta (m ^ 3) hsmall
  convert h using 1
  congr 2
  calc
    2 * ((Fintype.card V : ℝ) - 1) / (3 * m) =
        2 * (((Fintype.card V : ℝ) - 1) / (3 * m)) := by ring
    _ = _ := by rw [← hproduct]; ring

/-- Arbitrarily many expected singleton coordinates coexist with arbitrarily
small variation error: taking `k=m³` and `beta=m⁻²` leaves `k*beta=m`.
The bound involves only the fixed alphabet and requested error. -/
theorem exists_cubeSlope_totalVariation_le (η : ℝ) (hη : 0 < η) (N : ℕ) :
    ∃ m : ℕ, N ≤ m ∧ 0 < m ∧
      totalVariation
        (independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3))
        (uniformWeights (Fin (m ^ 3) → V × V)) ≤ η := by
  obtain ⟨m, hm⟩ := exists_nat_ge
    (max (max (1 : ℝ) N)
      (max (2 * ((Fintype.card V : ℝ) - 1) / 3)
        (((Fintype.card V : ℝ) - 1) / (6 * η ^ 2))))
  have hlow : max (1 : ℝ) N ≤ m := (le_max_left _ _).trans hm
  have hmone : (1 : ℝ) ≤ m := (le_max_left _ _).trans hlow
  have hmpos : (0 : ℝ) < m := lt_of_lt_of_le (by norm_num) hmone
  have hN : (N : ℝ) ≤ m := (le_max_right _ _).trans hlow
  have hhigh : max (2 * ((Fintype.card V : ℝ) - 1) / 3)
      (((Fintype.card V : ℝ) - 1) / (6 * η ^ 2)) ≤ m :=
    (le_max_right _ _).trans hm
  have hsize := (le_max_left _ _).trans hhigh
  have hrate := (le_max_right _ _).trans hhigh
  have hrate' : (Fintype.card V : ℝ) - 1 ≤ (m : ℝ) * (6 * η ^ 2) :=
    (div_le_iff₀ (by positivity : (0 : ℝ) < 6 * η ^ 2)).mp hrate
  refine ⟨m, by exact_mod_cast hN, Nat.cast_pos.mp hmpos, ?_⟩
  refine (cubeSlope_totalVariation V m (Nat.cast_pos.mp hmpos) hsize).trans ?_
  apply (div_le_iff₀ (by norm_num : (0 : ℝ) < 2)).2
  apply (Real.sqrt_le_left (by positivity : 0 ≤ η * 2)).2
  apply (div_le_iff₀ (by positivity : (0 : ℝ) < 3 * m)).2
  nlinarith

theorem cube_expected_singletons (m : ℕ) (hm : 0 < m) :
    ((m ^ 3 : ℕ) : ℝ) * (1 / (m : ℝ) ^ 2) = m := by
  have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
  push_cast
  field_simp [hm']

end Singleton

variable {V : Type*} [Fintype V] [AddGroup V] [DecidableEq V]

omit [Fintype α] [DecidableEq V] in
/-- Pulling back an independent uniform intercept through any translation,
even one depending on all slopes and projection choices, leaves its joint law
unchanged. No independence of that translation from the slopes is assumed. -/
theorem translated_uniform_intercept (p : α → ℝ) (shift : α → V) (a : α) (z : V) :
    p a * uniformWeights V (z - shift a) = p a * uniformWeights V z := rfl

theorem uniform_intercept_pushforward (shift z : V) :
    (∑ t : V, if t + shift = z then uniformWeights V t else 0) =
      uniformWeights V z := by
  classical
  simp [← eq_sub_iff_add_eq, uniformWeights]

omit [Fintype α] in
theorem joint_intercept_pushforward (p : α → ℝ) (shift : α → V) (a : α) (z : V) :
    (∑ t : V, if t + shift a = z then p a * uniformWeights V t else 0) =
      p a * uniformWeights V z := by
  classical
  calc
    _ = p a * (∑ t : V, if t + shift a = z then uniformWeights V t else 0) := by
      rw [Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro t _
      split <;> simp_all
    _ = _ := by rw [uniform_intercept_pushforward]

omit [DecidableEq V] in
theorem totalVariation_append_uniform_intercept (p q : α → ℝ) :
    totalVariation (fun az : α × V => p az.1 * uniformWeights V az.2)
      (fun az : α × V => q az.1 * uniformWeights V az.2) = totalVariation p q := by
  exact MaxCutGames.Foundations.Repetition.totalVariation_append_kernel p q
    (fun _ => uniformWeights V) (fun _ => uniformWeights_isProbability)

end
end MaxCutGames.Decoder.SparseLaw

end OAI

	import Mathlib
	import OAI.Computability.MaxCut.PCP.Variation

	namespace OAI

	/-!
	The finite calculation behind v2 Lemma `projected-tv`. The three singleton
	lines are averaged before the divergence is taken; projection choices are
	not part of the compared sample space. Coordinates are independently sampled
	occurrences, irrespective of whether their equation IDs coincide.
	-/

	namespace MaxCutGames.Decoder.SparseLaw

	open scoped BigOperators
	open MaxCutGames.Foundations.Information

	noncomputable section

	variable {α : Type*} [Fintype α]

	def uniformWeights (α : Type*) [Fintype α] : α → ℝ :=
	fun _ => 1 / Fintype.card α

	theorem uniformWeights_pos [Nonempty α] (a : α) : 0 < uniformWeights α a := by
	exact div_pos (by norm_num) (Nat.cast_pos.mpr Fintype.card_pos)

	theorem uniformWeights_isProbability [Nonempty α] :
	IsProbability (uniformWeights α) := by
	constructor
	· exact fun a => (uniformWeights_pos a).le
	· simp [uniformWeights, Fintype.card_ne_zero]

	/-- Pearson divergence against the uniform law, written with its constant
	denominator cleared. `uniformChiSquare_eq_pearson` identifies the actual
	likelihood-ratio definition. -/
	def uniformChiSquare (p : α → ℝ) : ℝ :=
	(Fintype.card α : ℝ) * ∑ a, (p a - uniformWeights α a) ^ 2

	theorem uniformChiSquare_eq_pearson [Nonempty α] (p : α → ℝ) :
	uniformChiSquare p =
	∑ a, uniformWeights α a * (p a / uniformWeights α a - 1) ^ 2 := by
	rw [uniformChiSquare, Finset.mul_sum]
	apply Finset.sum_congr rfl
	intro a _
	have hn : (Fintype.card α : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
	simp only [uniformWeights]
	field_simp [hn]

	theorem uniformChiSquare_eq_secondMoment [Nonempty α] (p : α → ℝ)
	(hp : IsProbability p) :
	uniformChiSquare p = (Fintype.card α : ℝ) * (∑ a, p a ^ 2) - 1 := by
	have hn : (Fintype.card α : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
	have hpoint (a : α) :
	(Fintype.card α : ℝ) * (p a - uniformWeights α a) ^ 2 =
	(Fintype.card α : ℝ) * p a ^ 2 - 2 * p a + uniformWeights α a := by
	simp only [uniformWeights]
	field_simp [hn]
	ring
	rw [uniformChiSquare, Finset.mul_sum]
	simp only [hpoint, Finset.sum_add_distrib,
	Finset.sum_sub_distrib, ← Finset.mul_sum, hp.2,
	(uniformWeights_isProbability (α := α)).2]
	ring

	theorem uniformChiSquare_nonneg (p : α → ℝ) : 0 ≤ uniformChiSquare p := by
	unfold uniformChiSquare
	positivity

	def mixture (β : ℝ) (p : α → ℝ) : α → ℝ :=
	fun a => (1 - β) * uniformWeights α a + β * p a

	theorem mixture_isProbability [Nonempty α] (p : α → ℝ) (hp : IsProbability p)
	{β : ℝ} (hβ : 0 ≤ β) (hβ' : β ≤ 1) : IsProbability (mixture β p) := by
	constructor
	· intro a
	exact add_nonneg (mul_nonneg (sub_nonneg.mpr hβ') (uniformWeights_pos a).le)
	(mul_nonneg hβ (hp.1 a))
	· simp only [mixture, Finset.sum_add_distrib, ← Finset.mul_sum, hp.2,
	(uniformWeights_isProbability (α := α)).2]
	ring

	theorem uniformChiSquare_mixture (p : α → ℝ) (β : ℝ) :
	uniformChiSquare (mixture β p) = β ^ 2 * uniformChiSquare p := by
	have hpoint (a : α) :
	(mixture β p a - uniformWeights α a) ^ 2 =
	β ^ 2 * (p a - uniformWeights α a) ^ 2 := by
	unfold mixture
	ring
	simp only [uniformChiSquare, hpoint, ← Finset.mul_sum]
	ring

	theorem totalVariation_le_half_sqrt (p : α → ℝ) :
	totalVariation p (uniformWeights α) ≤ Real.sqrt (uniformChiSquare p) / 2 := by
	have hcs := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset α)
	(fun _ => (1 : ℝ)) (fun a => \|p a - uniformWeights α a\|)
	simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
	mul_one, sq_abs] at hcs
	have hs := Real.le_sqrt_of_sq_le hcs
	exact div_le_div_of_nonneg_right hs (by norm_num : (0 : ℝ) ≤ 2)

	def independentWeights (p : α → ℝ) (k : ℕ) : (Fin k → α) → ℝ :=
	fun x => ∏ i, p (x i)

	theorem independentWeights_isProbability (p : α → ℝ) (hp : IsProbability p)
	(k : ℕ) : IsProbability (independentWeights p k) := by
	constructor
	· intro x
	exact Finset.prod_nonneg fun i _ => hp.1 (x i)
	· change (∑ x : Fin k → α, ∏ i, p (x i)) = 1
	rw [← Fintype.sum_pow, hp.2, one_pow]

	theorem uniformChiSquare_independent [Nonempty α] (p : α → ℝ)
	(hp : IsProbability p) (k : ℕ) :
	uniformChiSquare (independentWeights p k) = (1 + uniformChiSquare p) ^ k - 1 := by
	rw [uniformChiSquare_eq_secondMoment _ (independentWeights_isProbability p hp k),
	uniformChiSquare_eq_secondMoment p hp]
	have hsecond : (∑ x : Fin k → α, independentWeights p k x ^ 2) =
	(∑ a, p a ^ 2) ^ k := by
	simp only [independentWeights, ← Finset.prod_pow]
	exact (Fintype.sum_pow (fun a => p a ^ 2) k).symm
	rw [hsecond]
	simp only [Fintype.card_fun, Fintype.card_fin, Nat.cast_pow]
	rw [← mul_pow]
	exact congrArg (fun t : ℝ => t ^ k - 1) (by ring)

	/-- A finite bound sufficient for the cube asymptotic, without introducing a
	limit theorem or choosing the alphabet after the repetition count. -/
	theorem pow_mul_one_sub_le_one (x : ℝ) (hx : 0 ≤ x) (k : ℕ) :
	(1 + x) ^ k * (1 - k * x) ≤ 1 := by
	induction k with
	\| zero => simp
	\| succ k ih =>
	have hp : 0 ≤ (1 + x) ^ k := pow_nonneg (by linarith) _
	have herr : 0 ≤ (1 + x) ^ k * ((k : ℝ) + 1) * x ^ 2 :=
	mul_nonneg (mul_nonneg hp (by positivity)) (sq_nonneg x)
	rw [pow_succ, Nat.cast_add, Nat.cast_one]
	nlinarith

	theorem pow_sub_one_le_fraction (x : ℝ) (hx : 0 ≤ x) (k : ℕ)
	(hk : (k : ℝ) * x < 1) :
	(1 + x) ^ k - 1 ≤ (k : ℝ) * x / (1 - (k : ℝ) * x) := by
	apply (le_div_iff₀ (sub_pos.mpr hk)).2
	have h := pow_mul_one_sub_le_one x hx k
	nlinarith

	theorem pow_sub_one_le_twice (x : ℝ) (hx : 0 ≤ x) (k : ℕ)
	(hk : (k : ℝ) * x ≤ 1 / 2) :
	(1 + x) ^ k - 1 ≤ 2 * (k : ℝ) * x := by
	have hk' : (k : ℝ) * x < 1 := by linarith
	refine (pow_sub_one_le_fraction x hx k hk').trans ?_
	apply (div_le_iff₀ (sub_pos.mpr hk')).2
	have hn : 0 ≤ (k : ℝ) * x := mul_nonneg (Nat.cast_nonneg _) hx
	nlinarith

	section Singleton

	variable (V : Type*) [Fintype V] [Zero V] [DecidableEq V]

	/-- At the origin all three singleton lines contribute. -/
	def lineCount (z : V × V) : ℝ :=
	(if z.2 = 0 then 1 else 0) + (if z.1 = 0 then 1 else 0) +
	(if z.1 = z.2 then 1 else 0)

	/-- The average of the laws of `(a,0)`, `(0,a)`, and `(a,a)` for uniform `a`. -/
	def singletonPairWeights (z : V × V) : ℝ := lineCount V z / (3 * Fintype.card V)

	/-- Explicit finite pushforward of the three equally likely singleton choices.
	The same uniform slope supplies both entries of the diagonal choice. -/
	def singletonSamplerWeights (z : V × V) : ℝ :=
	((∑ a : V, if (a, 0) = z then uniformWeights V a else 0) +
	(∑ a : V, if (0, a) = z then uniformWeights V a else 0) +
	(∑ a : V, if (a, a) = z then uniformWeights V a else 0)) / 3

	theorem singletonSamplerWeights_eq (z : V × V) :
	singletonSamplerWeights V z = singletonPairWeights V z := by
	classical
	rcases z with ⟨x, y⟩
	have hleft : (∑ a : V, if (a, 0) = (x, y) then uniformWeights V a else 0) =
	if y = 0 then 1 / (Fintype.card V : ℝ) else 0 := by
	by_cases hy : y = 0
	· simp [hy, uniformWeights]
	· simp [Prod.mk.injEq, Ne.symm hy, hy]
	have hright : (∑ a : V, if (0, a) = (x, y) then uniformWeights V a else 0) =
	if x = 0 then 1 / (Fintype.card V : ℝ) else 0 := by
	by_cases hx : x = 0
	· simp [hx, uniformWeights]
	· simp [Prod.mk.injEq, Ne.symm hx, hx]
	have hdiag : (∑ a : V, if (a, a) = (x, y) then uniformWeights V a else 0) =
	if x = y then 1 / (Fintype.card V : ℝ) else 0 := by
	by_cases hxy : x = y
	· simp [hxy, uniformWeights]
	· have hnone (a : V) : (a, a) ≠ (x, y) := by
	intro ha
	have he := Prod.mk.inj ha
	exact hxy (he.1.symm.trans he.2)
	simp [hnone, hxy]
	simp only [singletonSamplerWeights, hleft, hright, hdiag,
	singletonPairWeights, lineCount]
	split_ifs <;> ring

	theorem lineCount_sum : ∑ z : V × V, lineCount V z = 3 * Fintype.card V := by
	have h₁ : (∑ z : V × V, if z.2 = 0 then (1 : ℝ) else 0) = Fintype.card V := by
	rw [Fintype.sum_prod_type]
	simp
	have h₂ : (∑ z : V × V, if z.1 = 0 then (1 : ℝ) else 0) = Fintype.card V := by
	rw [Fintype.sum_prod_type, Finset.sum_comm]
	simp
	have h₃ : (∑ z : V × V, if z.1 = z.2 then (1 : ℝ) else 0) = Fintype.card V := by
	rw [Fintype.sum_prod_type]
	simp
	simp only [lineCount, Finset.sum_add_distrib, h₁, h₂, h₃]
	ring

	omit [Fintype V] in
	theorem lineCount_sq (z : V × V) :
	lineCount V z ^ 2 = lineCount V z + (if z.1 = 0 ∧ z.2 = 0 then 6 else 0) := by
	by_cases hx : z.1 = 0 <;> by_cases hy : z.2 = 0 <;>
	by_cases hxy : z.1 = z.2 <;> simp_all [lineCount] ; norm_num

	theorem lineCount_sq_sum :
	∑ z : V × V, lineCount V z ^ 2 = 3 * Fintype.card V + 6 := by
	simp only [lineCount_sq, Finset.sum_add_distrib, lineCount_sum]
	congr 1
	rw [Fintype.sum_prod_type]
	rw [Finset.sum_eq_single (0 : V)]
	· simp
	· intro x _ hx
	simp [hx]
	· simp

	theorem singletonPairWeights_isProbability : IsProbability (singletonPairWeights V) := by
	have hn : (0 : ℝ) < Fintype.card V := Nat.cast_pos.mpr Fintype.card_pos
	constructor
	· intro z
	unfold singletonPairWeights lineCount
	positivity
	· simp only [singletonPairWeights, div_eq_mul_inv, ← Finset.sum_mul, lineCount_sum]
	exact mul_inv_cancel₀ (ne_of_gt (mul_pos (by norm_num) hn))

	theorem singletonPairWeights_secondMoment :
	∑ z : V × V, singletonPairWeights V z ^ 2 =
	(3 * Fintype.card V + 6) / (3 * Fintype.card V) ^ 2 := by
	simp only [singletonPairWeights, div_pow]
	simp only [div_eq_mul_inv, ← Finset.sum_mul, lineCount_sq_sum]

	theorem singletonPairWeights_chiSquare :
	uniformChiSquare (singletonPairWeights V) = ((Fintype.card V : ℝ) - 1) / 3 := by
	rw [uniformChiSquare_eq_secondMoment _ (singletonPairWeights_isProbability V),
	singletonPairWeights_secondMoment]
	simp only [Fintype.card_prod, Nat.cast_mul]
	have hn : (Fintype.card V : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
	field_simp [hn]
	ring

	/-- Exact finite divergence of the padded slope law. -/
	theorem paddedSlope_chiSquare (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) (k : ℕ) :
	uniformChiSquare (independentWeights (mixture β (singletonPairWeights V)) k) =
	(1 + β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ^ k - 1 := by
	rw [uniformChiSquare_independent _
	(mixture_isProbability _ (singletonPairWeights_isProbability V) hβ hβ'),
	uniformChiSquare_mixture, singletonPairWeights_chiSquare]
	exact congrArg (fun t : ℝ => (1 + t) ^ k - 1) (by ring)

	theorem paddedSlope_totalVariation (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) (k : ℕ) :
	totalVariation (independentWeights (mixture β (singletonPairWeights V)) k)
	(uniformWeights (Fin k → V × V)) ≤
	Real.sqrt ((1 + β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ^ k - 1) / 2 := by
	simpa only [paddedSlope_chiSquare V β hβ hβ' k] using
	totalVariation_le_half_sqrt (independentWeights (mixture β (singletonPairWeights V)) k)

	theorem paddedSlope_totalVariation_small (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1)
	(k : ℕ)
	(hk : (k : ℝ) * (β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ≤ 1 / 2) :
	totalVariation (independentWeights (mixture β (singletonPairWeights V)) k)
	(uniformWeights (Fin k → V × V)) ≤
	Real.sqrt (2 * (k : ℝ) * (β ^ 2 * ((Fintype.card V : ℝ) - 1) / 3)) / 2 := by
	have hn : (1 : ℝ) ≤ Fintype.card V := by
	exact_mod_cast (Nat.succ_le_iff.mpr (Fintype.card_pos : 0 < Fintype.card V))
	refine (paddedSlope_totalVariation V β hβ hβ' k).trans ?_
	apply div_le_div_of_nonneg_right _ (by norm_num)
	apply Real.sqrt_le_sqrt
	exact pow_sub_one_le_twice _ (by positivity) _ hk

	theorem cubeSlope_totalVariation (m : ℕ) (hm : 0 < m)
	(hsize : 2 * ((Fintype.card V : ℝ) - 1) / 3 ≤ m) :
	totalVariation
	(independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3))
	(uniformWeights (Fin (m ^ 3) → V × V)) ≤
	Real.sqrt (2 * ((Fintype.card V : ℝ) - 1) / (3 * m)) / 2 := by
	have hmpos : (0 : ℝ) < m := Nat.cast_pos.mpr hm
	have hmone : (1 : ℝ) ≤ m := by exact_mod_cast (Nat.succ_le_iff.mpr hm)
	have hbeta : (1 : ℝ) / (m : ℝ) ^ 2 ≤ 1 := by
	apply (div_le_iff₀ (sq_pos_of_pos hmpos)).2
	nlinarith
	have hproduct : ((m ^ 3 : ℕ) : ℝ) *
	((1 / (m : ℝ) ^ 2) ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) =
	((Fintype.card V : ℝ) - 1) / (3 * m) := by
	push_cast
	field_simp [hmpos.ne']

	have hsmall : ((m ^ 3 : ℕ) : ℝ) *
	((1 / (m : ℝ) ^ 2) ^ 2 * ((Fintype.card V : ℝ) - 1) / 3) ≤ 1 / 2 := by
	rw [hproduct]
	apply (div_le_iff₀ (mul_pos (by norm_num) hmpos)).2
	linarith
	have h := paddedSlope_totalVariation_small V (1 / (m : ℝ) ^ 2)
	(by positivity) hbeta (m ^ 3) hsmall
	convert h using 1
	congr 2
	calc
	2 * ((Fintype.card V : ℝ) - 1) / (3 * m) =
	2 * (((Fintype.card V : ℝ) - 1) / (3 * m)) := by ring
	_ = _ := by rw [← hproduct]; ring

	/-- Arbitrarily many expected singleton coordinates coexist with arbitrarily
	small variation error: taking `k=m³` and `beta=m⁻²` leaves `k*beta=m`.
	The bound involves only the fixed alphabet and requested error. -/
	theorem exists_cubeSlope_totalVariation_le (η : ℝ) (hη : 0 < η) (N : ℕ) :
	∃ m : ℕ, N ≤ m ∧ 0 < m ∧
	totalVariation
	(independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3))
	(uniformWeights (Fin (m ^ 3) → V × V)) ≤ η := by
	obtain ⟨m, hm⟩ := exists_nat_ge
	(max (max (1 : ℝ) N)
	(max (2 * ((Fintype.card V : ℝ) - 1) / 3)
	(((Fintype.card V : ℝ) - 1) / (6 * η ^ 2))))
	have hlow : max (1 : ℝ) N ≤ m := (le_max_left _ _).trans hm
	have hmone : (1 : ℝ) ≤ m := (le_max_left _ _).trans hlow
	have hmpos : (0 : ℝ) < m := lt_of_lt_of_le (by norm_num) hmone
	have hN : (N : ℝ) ≤ m := (le_max_right _ _).trans hlow
	have hhigh : max (2 * ((Fintype.card V : ℝ) - 1) / 3)
	(((Fintype.card V : ℝ) - 1) / (6 * η ^ 2)) ≤ m :=
	(le_max_right _ _).trans hm
	have hsize := (le_max_left _ _).trans hhigh
	have hrate := (le_max_right _ _).trans hhigh
	have hrate' : (Fintype.card V : ℝ) - 1 ≤ (m : ℝ) * (6 * η ^ 2) :=
	(div_le_iff₀ (by positivity : (0 : ℝ) < 6 * η ^ 2)).mp hrate
	refine ⟨m, by exact_mod_cast hN, Nat.cast_pos.mp hmpos, ?_⟩
	refine (cubeSlope_totalVariation V m (Nat.cast_pos.mp hmpos) hsize).trans ?_
	apply (div_le_iff₀ (by norm_num : (0 : ℝ) < 2)).2
	apply (Real.sqrt_le_left (by positivity : 0 ≤ η * 2)).2
	apply (div_le_iff₀ (by positivity : (0 : ℝ) < 3 * m)).2
	nlinarith

	theorem cube_expected_singletons (m : ℕ) (hm : 0 < m) :
	((m ^ 3 : ℕ) : ℝ) * (1 / (m : ℝ) ^ 2) = m := by
	have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
	push_cast
	field_simp [hm']

	end Singleton

	variable {V : Type*} [Fintype V] [AddGroup V] [DecidableEq V]

	omit [Fintype α] [DecidableEq V] in
	/-- Pulling back an independent uniform intercept through any translation,
	even one depending on all slopes and projection choices, leaves its joint law
	unchanged. No independence of that translation from the slopes is assumed. -/
	theorem translated_uniform_intercept (p : α → ℝ) (shift : α → V) (a : α) (z : V) :
	p a * uniformWeights V (z - shift a) = p a * uniformWeights V z := rfl

	theorem uniform_intercept_pushforward (shift z : V) :
	(∑ t : V, if t + shift = z then uniformWeights V t else 0) =
	uniformWeights V z := by
	classical
	simp [← eq_sub_iff_add_eq, uniformWeights]

	omit [Fintype α] in
	theorem joint_intercept_pushforward (p : α → ℝ) (shift : α → V) (a : α) (z : V) :
	(∑ t : V, if t + shift a = z then p a * uniformWeights V t else 0) =
	p a * uniformWeights V z := by
	classical
	calc
	_ = p a * (∑ t : V, if t + shift a = z then uniformWeights V t else 0) := by
	rw [Finset.mul_sum]
	apply Finset.sum_congr rfl
	intro t _
	split <;> simp_all
	_ = _ := by rw [uniform_intercept_pushforward]

	omit [DecidableEq V] in
	theorem totalVariation_append_uniform_intercept (p q : α → ℝ) :
	totalVariation (fun az : α × V => p az.1 * uniformWeights V az.2)
	(fun az : α × V => q az.1 * uniformWeights V az.2) = totalVariation p q := by
	exact MaxCutGames.Foundations.Repetition.totalVariation_append_kernel p q
	(fun _ => uniformWeights V) (fun _ => uniformWeights_isProbability)

	end
	end MaxCutGames.Decoder.SparseLaw

	end OAI