| import Mathlib |
| import OAI.Computability.VertexCover.Repetition.CompletedSampling |
| import OAI.Computability.VertexCover.Repetition.GameConditioning |
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| namespace OAI |
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| namespace UniqueGames.Foundations.Repetition |
| open scoped BigOperators |
| open Games CorrelatedSampling |
| noncomputable section |
| variable {Q₁ Q₂ A₁ A₂ S : Type*} |
| [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] [Fintype S] |
| [Nonempty A₁] [Nonempty A₂] [Nonempty S] |
| [DecidableEq Q₁] [DecidableEq Q₂] [DecidableEq S] {n : Nat} |
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| theorem coordinate_probability_le_value_of_local_completion |
| (G : Game Q₁ Q₂ A₁ A₂) (j : Fin n) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (σ : FiniteDistribution ((Q₁ × Q₂) × S)) |
| (profileL : Q₁ → FiniteDistribution S) (profileR : Q₂ → FiniteDistribution S) |
| (completionL : Q₁ × S → FiniteDistribution (Fin n → Q₁)) |
| (completionR : Q₂ × S → FiniteDistribution (Fin n → Q₂)) |
| (supportL : ∀ q xs, (completionL q).weight xs ≠ 0 → xs j = q.1) |
| (supportR : ∀ q ys, (completionR q).weight ys ≠ 0 → ys j = q.1) |
| (fallback : S) : |
| (σ.mixture (fun z => (completionL (z.1.1,z.2)).product |
| (completionR (z.1.2,z.2)))).probability (G.coordinateWin strategy j) ≤ |
| G.value + |
| (2 * Information.totalVariation σ.weight |
| (fun z => G.questions.weight z.1 * (profileL z.1.1).weight z.2) + |
| 2 * Information.totalVariation σ.weight |
| (fun z => G.questions.weight z.1 * (profileR z.1.2).weight z.2)) := by |
| let targetLaw := σ.mixture (fun z => (completionL (z.1.1,z.2)).product (completionR (z.1.2,z.2))) |
| let δ := 2 * Information.totalVariation σ.weight |
| (fun z => G.questions.weight z.1 * (profileL z.1.1).weight z.2) + |
| 2 * Information.totalVariation σ.weight |
| (fun z => G.questions.weight z.1 * (profileR z.1.2).weight z.2) |
| change targetLaw.probability (G.coordinateWin strategy j) ≤ G.value + δ |
| by_contra h |
| have gap : 0 < targetLaw.probability (G.coordinateWin strategy j) - (G.value + δ) := |
| sub_pos.mpr (lt_of_not_ge h) |
| let η := (targetLaw.probability (G.coordinateWin strategy j) - (G.value + δ)) / 2 |
| have hη : 0 < η := by dsimp [η]; linarith |
| obtain ⟨N,hN⟩ := samplerOutputLaw_arbitrarily_close σ G.questions profileL profileR fallback η hη |
| let thresholds := sharedProfileThresholds profileL profileR |
| let γ := traceSeedLaw (rectangleDistribution (α := S) thresholds) N |
| let sL := samplerLocal thresholds profileL fallback N |
| let sR := samplerLocal thresholds profileR fallback N |
| have hvariation := CompletedSampling.coordinateOutputLaw_totalVariation_le |
| G.questions γ j sL sR completionL completionR supportL supportR σ |
| have hclose : (CompletedSampling.coordinateOutputLaw G.questions γ j sL sR completionL completionR).totalVariation targetLaw ≤ δ + η := |
| hvariation.trans hN |
| have hsuccess := G.coordinate_probability_le_value_add_embedding_distance j targetLaw |
| (CompletedSampling.seedLaw γ completionL completionR) |
| (CompletedSampling.coordinateLeft j sL) (CompletedSampling.coordinateRight j sR) |
| (CompletedSampling.coordinateLeft_preserves j sL) |
| (CompletedSampling.coordinateRight_preserves j sR) strategy |
| have hsym : targetLaw.totalVariation |
| (G.localEmbeddingLaw (CompletedSampling.seedLaw γ completionL completionR) |
| (CompletedSampling.coordinateLeft j sL) (CompletedSampling.coordinateRight j sR)) = |
| (CompletedSampling.coordinateOutputLaw G.questions γ j sL sR completionL completionR).totalVariation targetLaw := by |
| unfold FiniteDistribution.totalVariation |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro q _ |
| exact abs_sub_comm _ _ |
| rw [hsym] at hsuccess |
| dsimp [η] at hclose |
| linarith |
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| end |
| end UniqueGames.Foundations.Repetition |
| end |
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| 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 |
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| end OAI |
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