import Mathlib
import OAI.Computability.VertexCover.Games.Basic
import OAI.Computability.VertexCover.Information.Finite

namespace OAI

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namespace UniqueGames.Foundations.Games.FiniteDistribution

open scoped BigOperators
noncomputable section

variable {Ω : Type*} [Fintype Ω]

def totalVariation (μ ν : FiniteDistribution Ω) : ℝ :=
  (∑ x, |μ.weight x - ν.weight x|) / 2

theorem totalVariation_nonnegative (μ ν : FiniteDistribution Ω) :
    0 ≤ μ.totalVariation ν :=
  div_nonneg (Finset.sum_nonneg fun _ _ => abs_nonneg _) (by norm_num)

theorem totalVariation_comm (μ ν : FiniteDistribution Ω) :
    μ.totalVariation ν = ν.totalVariation μ := by
  unfold totalVariation
  congr 1
  apply Finset.sum_congr rfl
  intro x _
  exact abs_sub_comm _ _

theorem probability_sub_le_totalVariation (μ ν : FiniteDistribution Ω)
    (event : Ω → Bool) :
    μ.probability event - ν.probability event ≤ μ.totalVariation ν := by
  have pointwise (x : Ω) :
      2 * ((if event x then μ.weight x else 0) -
        (if event x then ν.weight x else 0)) ≤
      |μ.weight x - ν.weight x| + (μ.weight x - ν.weight x) := by
    have h₁ := le_abs_self (μ.weight x - ν.weight x)
    have h₂ := neg_le_abs (μ.weight x - ν.weight x)
    cases event x <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> linarith
  have summed := Finset.sum_le_sum (s := Finset.univ) fun x _ => pointwise x
  simp only [← Finset.mul_sum, Finset.sum_add_distrib, Finset.sum_sub_distrib,
    μ.normalized, ν.normalized, sub_self, add_zero] at summed
  unfold probability totalVariation
  linarith

theorem probability_abs_sub_le_totalVariation (μ ν : FiniteDistribution Ω)
    (event : Ω → Bool) :
    |μ.probability event - ν.probability event| ≤ μ.totalVariation ν := by
  apply abs_le.mpr
  constructor
  · have h := ν.probability_sub_le_totalVariation μ event
    rw [totalVariation_comm] at h
    linarith
  · exact μ.probability_sub_le_totalVariation ν event

theorem probability_le_add_totalVariation (μ ν : FiniteDistribution Ω)
    (event : Ω → Bool) :
    μ.probability event ≤ ν.probability event + μ.totalVariation ν := by
  have h := μ.probability_sub_le_totalVariation ν event
  linarith

end
end UniqueGames.Foundations.Games.FiniteDistribution

end


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end

end OAI