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import OAI.Computability.VertexCover.Repetition.CompletedSampling
import OAI.Computability.VertexCover.Repetition.GameConditioning
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}
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
end
end UniqueGames.Foundations.Repetition
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
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