| import Mathlib |
| import OAI.Computability.VertexCover.Games.Basic |
| import OAI.Computability.VertexCover.Information.Finite |
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| 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 |
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|
| end |
| end UniqueGames.Foundations.Games.FiniteDistribution |
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| end |
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| end |
| end |
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| end |
| end |
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| end OAI |
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|
