import Mathlib

namespace OAI

section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
                                                                                             
section

namespace UniqueGames.Foundations.CorrelatedSampling

def firstAccepted {σ : Type*} (accept : σ → Bool) : List σ → Option σ
  | [] => none
  | proposal :: rest =>
      if accept proposal then some proposal else firstAccepted accept rest

theorem first_union_common {σ : Type*} (left right : σ → Bool)
    (proposals : List σ) (proposal : σ)
    (hfirst : firstAccepted (fun s => left s || right s) proposals = some proposal)
    (hleft : left proposal = true) (hright : right proposal = true) :
    firstAccepted left proposals = some proposal ∧
      firstAccepted right proposals = some proposal := by
  induction proposals with
  | nil => simp [firstAccepted] at hfirst
  | cons head tail ih =>
      by_cases hl : left head = true
      · have heq : head = proposal := by
          simpa [firstAccepted, hl] using hfirst
        subst head
        simp [firstAccepted, hleft, hright]
      · by_cases hr : right head = true
        · have heq : head = proposal := by
            simpa [firstAccepted, hl, hr] using hfirst
          subst head
          exact False.elim (hl hleft)
        · have htail : firstAccepted (fun s => left s || right s) tail = some proposal := by
            simpa [firstAccepted, hl, hr] using hfirst
          simpa [firstAccepted, hl, hr] using ih htail

def thresholdSample {α : Type*} (mass : α → Nat) (proposals : List (α × Nat)) :
    Option α :=
  (firstAccepted (fun s => decide (s.2 < mass s.1)) proposals).map Prod.fst

theorem thresholdSample_agreement {α : Type*} (left right : α → Nat)
    (proposals : List (α × Nat)) (a : α) (height : Nat)
    (hfirst : firstAccepted
      (fun s => decide (s.2 < left s.1) || decide (s.2 < right s.1)) proposals =
      some (a, height))
    (hcommon : height < min (left a) (right a)) :
    thresholdSample left proposals = some a ∧
      thresholdSample right proposals = some a := by
  have h := first_union_common
    (fun s : α × Nat => decide (s.2 < left s.1))
    (fun s : α × Nat => decide (s.2 < right s.1)) proposals (a, height) hfirst
    (by simpa using (lt_min_iff.mp hcommon).1)
    (by simpa using (lt_min_iff.mp hcommon).2)
  simp [thresholdSample, h.1, h.2]

noncomputable section FiniteWeights

variable {α : Type*} [Fintype α]

def totalVariation (p q : α → ℝ) : ℝ := (∑ a, |p a - q a|) / 2

def overlapMass (p q : α → ℝ) : ℝ := ∑ a, min (p a) (q a)

def unionMass (p q : α → ℝ) : ℝ := ∑ a, max (p a) (q a)

theorem totalVariation_nonneg (p q : α → ℝ) : 0 ≤ totalVariation p q := by
  unfold totalVariation
  exact div_nonneg (Finset.sum_nonneg (fun _ _ => abs_nonneg _)) (by norm_num)

theorem overlapMass_nonneg (p q : α → ℝ) (hp : ∀ a, 0 ≤ p a)
    (hq : ∀ a, 0 ≤ q a) : 0 ≤ overlapMass p q := by
  exact Finset.sum_nonneg (fun a _ => le_min (hp a) (hq a))

private theorem min_abs_identity (x y : ℝ) : 2 * min x y + |x - y| = x + y := by
  rcases le_total x y with h | h
  · rw [min_eq_left h, abs_of_nonpos (sub_nonpos.mpr h)]
    ring
  · rw [min_eq_right h, abs_of_nonneg (sub_nonneg.mpr h)]
    ring

private theorem max_abs_identity (x y : ℝ) : 2 * max x y - |x - y| = x + y := by
  rcases le_total x y with h | h
  · rw [max_eq_right h, abs_of_nonpos (sub_nonpos.mpr h)]
    ring
  · rw [max_eq_left h, abs_of_nonneg (sub_nonneg.mpr h)]
    ring

theorem overlapMass_eq (p q : α → ℝ) (hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
    overlapMass p q = 1 - totalVariation p q := by
  have h := Finset.sum_congr (s₁ := Finset.univ) (s₂ := Finset.univ) rfl
    (fun a _ => min_abs_identity (p a) (q a))
  simp only [Finset.sum_add_distrib, ← Finset.mul_sum, hp, hq] at h
  unfold overlapMass totalVariation
  linarith

theorem unionMass_eq (p q : α → ℝ) (hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
    unionMass p q = 1 + totalVariation p q := by
  have h := Finset.sum_congr (s₁ := Finset.univ) (s₂ := Finset.univ) rfl
    (fun a _ => max_abs_identity (p a) (q a))
  simp only [Finset.sum_sub_distrib, Finset.sum_add_distrib, ← Finset.mul_sum, hp, hq] at h
  unfold unionMass totalVariation
  linarith

theorem totalVariation_le_one (p q : α → ℝ) (hp : ∀ a, 0 ≤ p a)
    (hq : ∀ a, 0 ≤ q a) (hpsum : ∑ a, p a = 1) (hqsum : ∑ a, q a = 1) :
    totalVariation p q ≤ 1 := by
  have h := overlapMass_nonneg p q hp hq
  rw [overlapMass_eq p q hpsum hqsum] at h
  linarith

theorem overlap_union_ratio (p q : α → ℝ)
    (hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
    overlapMass p q / unionMass p q =
      (1 - totalVariation p q) / (1 + totalVariation p q) := by
  rw [overlapMass_eq p q hp hq, unionMass_eq p q hp hq]

theorem overlap_union_ratio_lower_bound (p q : α → ℝ)
    (hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
    1 - 2 * totalVariation p q ≤ overlapMass p q / unionMass p q := by
  rw [overlap_union_ratio p q hp hq]
  have htv := totalVariation_nonneg p q
  have hden : 0 < 1 + totalVariation p q := by linarith
  apply (le_div_iff₀ hden).2
  nlinarith [sq_nonneg (totalVariation p q)]

theorem first_union_disagreement_upper_bound (p q : α → ℝ)
    (hp : ∑ a, p a = 1) (hq : ∑ a, q a = 1) :
    1 - overlapMass p q / unionMass p q ≤ 2 * totalVariation p q := by
  linarith [overlap_union_ratio_lower_bound p q hp hq]

end FiniteWeights

end UniqueGames.Foundations.CorrelatedSampling

end


end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI