lean/OAI/Combinatorics/IndependentSets/Expansion/PoweringDecoding.lean

import Mathlib

namespace OAI

namespace IndependentSetsGames.Foundations.PCP.PoweringDecoding

open scoped BigOperators

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

noncomputable def mass (opinion : Ω → A) (a : A) : ℝ := by
  classical
  exact Finset.univ.expect (fun ω => if opinion ω = a then 1 else 0)

omit [Nonempty A] in
theorem mass_sum (opinion : Ω → A) : (∑ a, mass opinion a) = 1 := by
  classical
  calc
    (∑ a, mass opinion a) =
        (Finset.univ : Finset Ω).expect
          (fun ω => ∑ a : A, if opinion ω = a then (1 : ℝ) else 0) :=
      (Finset.expect_sum_comm (Finset.univ : Finset Ω) (Finset.univ : Finset A)
        (fun ω a => if opinion ω = a then (1 : ℝ) else 0)).symm
    _ = (Finset.univ : Finset Ω).expect (fun _ => (1 : ℝ)) := by
      apply Finset.expect_congr rfl
      intro ω _
      simp
    _ = 1 := Finset.expect_const Finset.univ_nonempty 1

omit [Nonempty Ω] in
theorem exists_mode (opinion : Ω → A) :
    ∃ a : A, ∀ b : A, mass opinion b ≤ mass opinion a := by
  obtain ⟨a, _, ha⟩ := Finset.exists_max_image (Finset.univ : Finset A)
    (mass opinion) Finset.univ_nonempty
  exact ⟨a, fun b => ha b (Finset.mem_univ b)⟩

noncomputable def mode (opinion : Ω → A) : A := Classical.choose (exists_mode opinion)

omit [Nonempty Ω] in
theorem mode_max (opinion : Ω → A) (b : A) :
    mass opinion b ≤ mass opinion (mode opinion) :=
  Classical.choose_spec (exists_mode opinion) b

theorem mode_mass_lower (opinion : Ω → A) :
    1 / (Fintype.card A : ℝ) ≤ mass opinion (mode opinion) := by
  have hk : 0 < (Fintype.card A : ℝ) := Nat.cast_pos.mpr Fintype.card_pos
  have hsum : (1 : ℝ) ≤ mass opinion (mode opinion) * (Fintype.card A : ℝ) := by
    calc
      (1 : ℝ) = ∑ a : A, mass opinion a := (mass_sum opinion).symm
      _ ≤ ∑ _a : A, mass opinion (mode opinion) :=
        Finset.sum_le_sum (fun a _ => mode_max opinion a)
      _ = mass opinion (mode opinion) * (Fintype.card A : ℝ) := by simp [mul_comm]
  exact (div_le_iff₀ hk).2 hsum

theorem exists_letter_mass_lower (opinion : Ω → A) :
    ∃ a : A, 1 / (Fintype.card A : ℝ) ≤ mass opinion a :=
  ⟨mode opinion, mode_mass_lower opinion⟩

end IndependentSetsGames.Foundations.PCP.PoweringDecoding
noncomputable section

namespace IndependentSetsGames.Foundations.PCP.PoweringMoment

open scoped BigOperators

def bit (p : Prop) : ℝ := by
  classical
  exact if p then 1 else 0

def hits {I Ω : Type*} [Fintype I] (E : I → Ω → Prop) (ω : Ω) : ℝ :=
  ∑ i, bit (E i ω)

def mean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ := Finset.univ.expect f

theorem bit_nonneg (p : Prop) : 0 ≤ bit p := by
  classical
  by_cases hp : p <;> simp [bit, hp]

theorem bit_sq (p : Prop) : bit p ^ 2 = bit p := by
  classical
  by_cases hp : p <;> simp [bit, hp]

theorem bit_mul (p q : Prop) : bit p * bit q = bit (p ∧ q) := by
  classical
  by_cases hp : p <;> by_cases hq : q <;> simp [bit, hp, hq]

theorem bit_mono {p q : Prop} (hpq : p → q) : bit p ≤ bit q := by
  classical
  by_cases hp : p
  · simp [bit, hp, hpq hp]
  · have hz : bit p = 0 := by simp [bit, hp]
    rw [hz]
    exact bit_nonneg q

theorem hits_nonneg {I Ω : Type*} [Fintype I] (E : I → Ω → Prop) (ω : Ω) :
    0 ≤ hits E ω := Finset.sum_nonneg (fun i _ => bit_nonneg (E i ω))

theorem mean_nonneg {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (hf : ∀ ω, 0 ≤ f ω) :
    0 ≤ mean f := Finset.expect_nonneg (fun ω _ => hf ω)

theorem mean_hits {I Ω : Type*} [Fintype I] [Fintype Ω] (E : I → Ω → Prop) :
    mean (hits E) = ∑ i, mean (fun ω => bit (E i ω)) := by
  exact Finset.expect_sum_comm _ _ _

theorem second_moment_eq {I Ω : Type*} [Fintype I] [Fintype Ω]
    (E : I → Ω → Prop) :
    mean (fun ω => hits E ω ^ 2) =
      ∑ i, ∑ j, mean (fun ω => bit (E i ω ∧ E j ω)) := by
  simp only [hits, pow_two, Finset.sum_mul, Finset.mul_sum, bit_mul,
    mean, Finset.expect_sum_comm]
  exact Finset.sum_comm

theorem mean_witness_sq_le {I Ω : Type*} [Fintype I] [Fintype Ω]
    (E W : I → Ω → Prop) (reject : Ω → Prop)
    (hWE : ∀ i ω, W i ω → E i ω)
    (hWR : ∀ i ω, W i ω → reject ω) :
    mean (hits W) ^ 2 ≤
      mean (fun ω => bit (reject ω)) * mean (fun ω => hits E ω ^ 2) := by
  classical
  have hNH (ω : Ω) : hits W ω ≤ hits E ω :=
    Finset.sum_le_sum (fun i _ => bit_mono (hWE i ω))
  have hzero (ω : Ω) (hr : ¬ reject ω) : hits W ω = 0 := by
    apply Finset.sum_eq_zero
    intro i _
    have hw : ¬ W i ω := fun h => hr (hWR i ω h)
    simp [bit, hw]
  have hsupport (ω : Ω) : bit (reject ω) * hits W ω = hits W ω := by
    by_cases hr : reject ω
    · simp [bit, hr]
    · rw [hzero ω hr]
      simp
  have hCS := Finset.expect_mul_sq_le_sq_mul_sq Finset.univ
    (fun ω => bit (reject ω)) (hits W)
  have hsquares : mean (fun ω => hits W ω ^ 2) ≤ mean (fun ω => hits E ω ^ 2) := by
    apply Finset.expect_le_expect
    intro ω _
    simpa only [pow_two] using
      (mul_le_mul (hNH ω) (hNH ω) (hits_nonneg W ω) (hits_nonneg E ω))
  calc
    _ ≤ mean (fun ω => bit (reject ω)) * mean (fun ω => hits W ω ^ 2) := by
      simpa only [mean, hsupport, bit_sq] using hCS
    _ ≤ _ := mul_le_mul_of_nonneg_left hsquares
      (mean_nonneg _ (fun ω => bit_nonneg (reject ω)))

theorem rejection_lower_bound {I Ω : Type*} [Fintype I] [Fintype Ω]
    (E W : I → Ω → Prop) (reject : Ω → Prop)
    (hWE : ∀ i ω, W i ω → E i ω)
    (hWR : ∀ i ω, W i ω → reject ω)
    (a b : ℝ) (ha : 0 < a) (hb : 0 < b)
    (hfirst : a ≤ mean (hits W))
    (hsecond : mean (fun ω => hits E ω ^ 2) ≤ b) :
    a ^ 2 / b ≤ mean (fun ω => bit (reject ω)) := by
  have hN : 0 ≤ mean (hits W) := mean_nonneg _ (hits_nonneg W)
  have hρ : 0 ≤ mean (fun ω => bit (reject ω)) :=
    mean_nonneg _ (fun ω => bit_nonneg (reject ω))
  have haSq : a ^ 2 ≤ mean (hits W) ^ 2 := by
    simpa only [pow_two] using mul_le_mul hfirst hfirst ha.le hN
  have hCS := mean_witness_sq_le E W reject hWE hWR
  apply (div_le_iff₀ hb).2
  exact haSq.trans (hCS.trans (mul_le_mul_of_nonneg_left hsecond hρ))

end IndependentSetsGames.Foundations.PCP.PoweringMoment
end

end OAI