Erdős problem 579

/-
Upstream source notice

Selected upstream portions are modified from TCSlib
https://github.com/Shilun-Allan-Li/tcslib
commit a8c77c605cb9843b3eaebe11253c4eef15ded3f7, and FABL
https://github.com/Polarnova/FABL
commit 164f830312946ccac6e7fdc7bb9493f0149e820d.
Their original source notices are retained below. Changes include
the Lean 4.33.1/Mathlib port, selection of needed declarations,
expanded notation and explicit replacement of source automation.
The following Apache 2.0 license applies to those upstream portions.

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end Bounty

section

/- The reverse-mixing closure is copied as source; no TCSlib/FABL import is used. -/
open scoped BigOperators Topology Classical

namespace BooleanAnalysis

variable {n : ℕ}

-- TCSlib.BooleanAnalysis.Basic:62
/-- Defines the Boolean hypercube `{0,1}ⁿ`.

**Source:** [OD14, §1.1]. -/
abbrev BoolCube (n : ℕ) := Fin n → Bool

-- TCSlib.BooleanAnalysis.Basic:67
/-- Defines a real-valued Boolean function `f : {0,1}ⁿ → ℝ`.

**Source:** [OD14, §1.1]. -/
abbrev BooleanFunc (n : ℕ) := BoolCube n → ℝ

-- TCSlib.BooleanAnalysis.Basic:76
/-- Defines the uniform probability weight `2⁻ⁿ` on each point of `{0,1}ⁿ`.

**Source:** [OD14, §1.1]. -/
noncomputable def uniformWeight (n : ℕ) : ℝ := (2 : ℝ)⁻¹ ^ n

-- TCSlib.BooleanAnalysis.Basic:81
/-- Expectation of `f` under the uniform measure on `{0,1}ⁿ`.
  `𝔼[f] = 2⁻ⁿ · ∑_{x ∈ {0,1}ⁿ} f(x)`.

**Source:** [OD14, §1.1]. -/
noncomputable def expect (f : BooleanFunc n) : ℝ :=
  uniformWeight n * ∑ x : BoolCube n, f x

-- TCSlib.BooleanAnalysis.Basic:88
/-- The Boolean-cube expectation is Mathlib's uniform expectation on the finite cube.

**Source:** [OD14, §1.1]; Mathlib, `Fintype.expect_eq_sum_div_card`. -/
lemma expect_eq_fintypeExpect (f : BooleanFunc n) : expect f = 𝔼 x, f x := by
  rw [Fintype.expect_eq_sum_div_card]
  unfold expect uniformWeight
  simp [Fintype.card_pi, Fintype.card_bool, div_eq_inv_mul]

-- TCSlib.BooleanAnalysis.Basic:96
/-- The `L²` inner product on Boolean functions with respect to the uniform measure:
  `⟪f, g⟫ = 𝔼[f · g] = 2⁻ⁿ · ∑_x f(x) g(x)`.

**Source:** [OD14, §1.2]. -/
noncomputable def innerProduct (f g : BooleanFunc n) : ℝ :=
  expect (fun x ↦ f x * g x)

-- TCSlib.BooleanAnalysis.Basic:138
/-- Converts a `Bool` to `{-1, 1} ⊆ ℝ`: `false ↦ 1`, `true ↦ -1`.

**Source:** [OD14, §1.1]. -/
def boolToSign (b : Bool) : ℝ := if b then -1 else 1

lemma boolToSign_false : boolToSign false = 1 := rfl
lemma boolToSign_true : boolToSign true = -1 := rfl

-- TCSlib.BooleanAnalysis.Basic:149
lemma boolToSign_sq (b : Bool) : boolToSign b ^ 2 = 1 := by
  cases b <;> simp [boolToSign]

-- TCSlib.BooleanAnalysis.Basic:153
lemma boolToSign_mul_self (b : Bool) : boolToSign b * boolToSign b = 1 := by
  cases b <;> simp [boolToSign]

-- TCSlib.BooleanAnalysis.Basic:157
/-- The Walsh–Fourier character `χ_S : {0,1}ⁿ → ℝ` associated to a set `S ⊆ [n]`.
  `χ_S(x) = ∏_{i ∈ S} (-1)^{x_i}`.

  This forms an orthonormal basis for `L²({0,1}ⁿ, uniform)`.

**Source:** [OD14, §1.2]. -/
noncomputable def chiS (S : Finset (Fin n)) : BooleanFunc n :=
  fun x ↦ ∏ i ∈ S, boolToSign (x i)

-- TCSlib.BooleanAnalysis.Basic:168
/-- The character `χ_∅` is the constant function `1`. -/
lemma chiS_empty : chiS ((∅ : Finset (Fin n))) = fun _ ↦ 1 := by
  ext x; simp [chiS]

-- TCSlib.BooleanAnalysis.Basic:173
/-- The character `χ_{i}` for a singleton `{i}` equals `(-1)^{x_i}`. -/
lemma chiS_singleton (i : Fin n) (x : BoolCube n) :
    chiS ({i}) x = boolToSign (x i) := by
  simp [chiS]

-- TCSlib.BooleanAnalysis.Basic:179
/-- Walsh characters take values in `{-1, 1}`. -/
lemma chiS_sq_eq_one (S : Finset (Fin n)) (x : BoolCube n) :
    chiS (S) x ^ 2 = 1 := by
  simp only [chiS]
  induction S using Finset.induction with
  | empty => simp
  | insert a s ha ih =>
    rw [Finset.prod_insert ha, mul_pow, boolToSign_sq, one_mul]
    exact ih

-- TCSlib.BooleanAnalysis.Basic:195
/-- The pointwise product of two Walsh characters is another Walsh character (up to sign),
  specifically `χ_S · χ_T = χ_{S Δ T}` where `Δ` denotes symmetric difference. -/
lemma chiS_mul_chiS (S T : Finset (Fin n)) (x : BoolCube n) :
    chiS (S) x * chiS (T) x = chiS (symmDiff S T) x := by
  simp only [chiS]
  -- Decompose: S = (S \ T) ∪ (S ∩ T), T = (T \ S) ∪ (T ∩ S)
  have hS : ∏ i ∈ S, boolToSign (x i) =
      (∏ i ∈ S \ T, boolToSign (x i)) * ∏ i ∈ S ∩ T, boolToSign (x i) := by
    conv_lhs => rw [← Finset.sdiff_union_inter S T]
    apply Finset.prod_union
    simp only [Finset.disjoint_left, Finset.mem_sdiff, Finset.mem_inter, not_and]
    tauto
  have hT : ∏ i ∈ T, boolToSign (x i) =
      (∏ i ∈ T \ S, boolToSign (x i)) * ∏ i ∈ S ∩ T, boolToSign (x i) := by
    conv_lhs => rw [← Finset.sdiff_union_inter T S]
    rw [Finset.inter_comm T S]
    apply Finset.prod_union
    simp only [Finset.disjoint_left, Finset.mem_sdiff, Finset.mem_inter, not_and]
    tauto
  -- The intersection product squares to 1
  have hcancel : (∏ i ∈ S ∩ T, boolToSign (x i)) * ∏ i ∈ S ∩ T, boolToSign (x i) = 1 := by
    rw [← Finset.prod_mul_distrib]; simp [boolToSign_mul_self]
  rw [hS, hT, symmDiff_def, Finset.sup_eq_union, Finset.prod_union disjoint_sdiff_sdiff]
  -- Goal: (A * P) * (B * P) = A * B  where P² = 1
  set P := ∏ i ∈ S ∩ T, boolToSign (x i)
  set A := ∏ i ∈ S \ T, boolToSign (x i)
  set B := ∏ i ∈ T \ S, boolToSign (x i)
  calc A * P * (B * P) = A * B * (P * P) := by ring
    _ = A * B * 1 := by rw [hcancel]
    _ = A * B := by ring

-- TCSlib.BooleanAnalysis.Basic:228
/-- The Fourier–Walsh coefficient of `f` at frequency `S`:
  `f̂(S) = ⟪f, χ_S⟫ = 2⁻ⁿ · ∑_x f(x) · χ_S(x)`.

**Source:** [OD14, §1.2]. -/
noncomputable def fourierCoeff (f : BooleanFunc n) (S : Finset (Fin n)) : ℝ :=
  innerProduct f (chiS S)

-- TCSlib.BooleanAnalysis.Basic:244
/-- Key identity: `∑_{S ⊆ [n]} ∏_{i∈S} c_i = ∏_i (1 + c_i)`.
  Used via `Finset.prod_one_add`. -/
private lemma sum_prod_subset_eq_prod_one_add (c : Fin n → ℝ) :
    ∑ S : Finset (Fin n), ∏ i ∈ S, c i =
    ∏ i : Fin n, (1 + c i) := by
  -- Use Finset.prod_one_add: ∏_{i∈s} (1 + f i) = ∑_{t∈s.powerset} ∏_{i∈t} f i
  rw [Finset.prod_one_add Finset.univ]
  -- Now RHS = ∑ t ∈ Finset.univ.powerset, ∏ i ∈ t, c i
  -- Reindex: Finset.univ.powerset ≅ all Finset (Fin n) via id
  apply Finset.sum_nbij id
  · intro t _; exact Finset.mem_powerset.mpr (Finset.subset_univ t)
  · intro t₁ _ t₂ _ h; exact h
  · intro t ht; exact ⟨t, Finset.mem_univ t, rfl⟩
  · intro t _; rfl

-- TCSlib.BooleanAnalysis.Basic:259
/-- The sum of `χ_S(x) * χ_S(y)` over all `S ⊆ [n]` equals `2ⁿ` if `x = y`, else `0`.
  This is the completeness kernel for the Walsh basis. -/
private lemma sum_chiS_mul_eq (x y : BoolCube n) :
    ∑ S : Finset (Fin n), chiS S x * chiS S y = if x = y then (2 : ℝ) ^ n else 0 := by
  simp only [chiS, ← Finset.prod_mul_distrib]
  rw [sum_prod_subset_eq_prod_one_add]
  split_ifs with hxy
  · subst hxy; simp only [boolToSign_mul_self]
    simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin]
    norm_num
  · obtain ⟨i, hi⟩ := Function.ne_iff.mp hxy
    apply Finset.prod_eq_zero (Finset.mem_univ i)
    have : boolToSign (x i) * boolToSign (y i) = -1 := by
      cases hxi : x i <;> cases hyi : y i <;> simp_all [boolToSign]
    simp [this]

-- TCSlib.BooleanAnalysis.Basic:275
/-- **Walsh Expansion**: every Boolean function `f : {0,1}ⁿ → ℝ` can be written as
  `f(x) = ∑_{S ⊆ [n]} f̂(S) · χ_S(x)`.

  This is the Fourier inversion formula for the uniform measure on `{0,1}ⁿ`.

**Source:** [OD14, §1.3]. -/
theorem walsh_expansion (f : BooleanFunc n) (x : BoolCube n) :
    f x = ∑ S : Finset (Fin n), fourierCoeff f S * chiS S x := by
  simp only [fourierCoeff, innerProduct, expect, uniformWeight]
  -- Goal: f x = ∑_S (2⁻ⁿ * ∑_y f(y) * χ_S(y)) * χ_S(x)
  -- Proof: show both sides equal 2⁻ⁿ * ∑_y f(y) * ∑_S χ_S(y) * χ_S(x)
  --        then use the completeness kernel
  symm
  calc ∑ S : Finset (Fin n), ((2:ℝ)⁻¹^n * ∑ y, f y * chiS S y) * chiS S x
      = (2:ℝ)⁻¹^n * ∑ y : BoolCube n, ∑ S : Finset (Fin n), f y * (chiS S y * chiS S x) := by
        -- Move 2⁻¹^n outside by rearranging: ∑_S (a * b_S) * c_S = a * ∑_S b_S * c_S,
        -- then swap sum order and distribute f y
        have step1 : ∑ S : Finset (Fin n), ((2:ℝ)⁻¹^n * ∑ y, f y * chiS S y) * chiS S x =
            (2:ℝ)⁻¹^n * ∑ S : Finset (Fin n), (∑ y, f y * chiS S y) * chiS S x := by
          rw [Finset.mul_sum]
          apply Finset.sum_congr rfl; intro S _; ring
        have step2 : ∑ S : Finset (Fin n), (∑ y, f y * chiS S y) * chiS S x =
            ∑ y : BoolCube n, ∑ S : Finset (Fin n), f y * (chiS S y * chiS S x) := by
          simp_rw [Finset.sum_mul]
          rw [Finset.sum_comm]
          apply Finset.sum_congr rfl; intro y _
          apply Finset.sum_congr rfl; intro S _; ring
        rw [step1, step2]
    _ = (2:ℝ)⁻¹^n * ∑ y : BoolCube n, f y * (∑ S, chiS S y * chiS S x) := by
        congr 1
        apply Finset.sum_congr rfl; intro y _
        rw [← Finset.mul_sum]
    _ = (2:ℝ)⁻¹^n * ∑ y : BoolCube n, f y * (if y = x then (2:ℝ)^n else 0) := by
        simp_rw [sum_chiS_mul_eq]
    _ = (2:ℝ)⁻¹^n * (f x * (2:ℝ)^n) := by
        congr 1
        simp [Finset.sum_ite_eq', Finset.mem_univ]
    _ = f x := by
        rw [← mul_assoc, mul_comm ((2:ℝ)⁻¹^n) (f x), mul_assoc, ← mul_pow,
            inv_mul_cancel₀ (by norm_num : (2:ℝ) ≠ 0), one_pow, mul_one]

-- TCSlib.BooleanAnalysis.Basic:333
/-- Summing `χ_S` over the entire hypercube gives `2ⁿ` if `S = ∅`, else `0`. -/
private lemma sum_chiS (S : Finset (Fin n)) :
    ∑ x : BoolCube n, chiS S x = if S = ∅ then 2 ^ n else 0 := by
  simp only [chiS]
  by_cases hS : S = ∅
  · subst hS; simp [Fintype.card_pi, Fintype.card_bool]
  · simp only [hS, if_false]
    have factored : ∑ x : BoolCube n, ∏ i ∈ S, boolToSign (x i) =
        ∑ x : BoolCube n, ∏ i : Fin n, (if i ∈ S then boolToSign (x i) else 1) := by
      congr 1; ext x; rw [← Finset.prod_filter]; simp
    rw [factored]
    -- Goal: ∑ x : BoolCube n, ∏ i : Fin n, g i (x i) = 0
    -- where g i b = if i ∈ S then boolToSign b else 1
    -- Factor: = ∏ i : Fin n, ∑ b : Bool, g i b  (by Fintype.prod_sum reversed)
    rw [show ∑ x : BoolCube n, ∏ i : Fin n, (if i ∈ S then boolToSign (x i) else 1) =
        ∏ i : Fin n, ∑ b : Bool, (if i ∈ S then boolToSign b else 1) from
      (Fintype.prod_sum (fun i b => if i ∈ S then boolToSign b else 1)).symm]
    obtain ⟨i, hi⟩ := Finset.nonempty_iff_ne_empty.mpr hS
    apply Finset.prod_eq_zero (Finset.mem_univ i)
    simp [hi, boolToSign]

-- TCSlib.BooleanAnalysis.Basic:354
/-- **Orthonormality**: `⟪χ_S, χ_T⟫ = [S = T]`.

  The Walsh characters form an orthonormal system in `L²({0,1}ⁿ, uniform)`.

**Source:** [OD14, §1.2]. -/
theorem fourier_coeff_chi (S T : Finset (Fin n)) :
    innerProduct (chiS S) (chiS T) = if S = T then 1 else 0 := by
  simp only [innerProduct, expect, uniformWeight]
  have step : ∑ x : BoolCube n, chiS S x * chiS T x =
      ∑ x : BoolCube n, chiS (symmDiff S T) x := by
    congr 1; ext x; exact chiS_mul_chiS S T x
  rw [step, sum_chiS]
  by_cases hst : S = T
  · -- S = T: symmDiff S T = ∅
    subst hst
    simp only [symmDiff_self, Finset.bot_eq_empty, ↓reduceIte]
    rw [← mul_pow]; norm_num
  · -- S ≠ T: symmDiff S T ≠ ∅
    have hd : symmDiff S T ≠ ∅ := by
      intro h
      apply hst
      have : symmDiff S T = ⊥ := by rwa [Finset.bot_eq_empty]
      exact symmDiff_eq_bot.mp this
    simp [hd, hst]

-- TCSlib.BooleanAnalysis.Basic:385
/-- **Parseval's Identity**: `‖f‖² = ∑_{S ⊆ [n]} f̂(S)²`.

  The sum of squared Fourier coefficients equals the squared `L²` norm.

**Source:** [OD14, §1.4]. -/
theorem parseval (f : BooleanFunc n) :
    innerProduct f f = ∑ S : Finset (Fin n), fourierCoeff f S ^ 2 := by
  -- Expand f = ∑_S f̂(S) χ_S and use bilinearity + orthonormality
  have expand : innerProduct f f =
      ∑ S : Finset (Fin n), ∑ T : Finset (Fin n),
        fourierCoeff f S * fourierCoeff f T * innerProduct (chiS S) (chiS T) := by
    -- Expand innerProduct and uniformWeight first so f x * f x becomes visible
    simp_rw [innerProduct, expect, uniformWeight]
    -- Now rewrite f(x)*f(x) using walsh_expansion
    simp_rw [show ∀ x : BoolCube n, f x * f x =
        (∑ S : Finset (Fin n), fourierCoeff f S * chiS S x) *
        (∑ T : Finset (Fin n), fourierCoeff f T * chiS T x) from fun x => by