| import OAI.Probability.EntangledGames.StrategyBridge |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ComplexOrder |
| namespace ThresholdParallelRepetition |
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| theorem threshold_parallel_repetition : |
| ∃ κ₀ : ℝ, 0 < κ₀ ∧ |
| ∀ (x y a b : ℕ) (G : Game x y a b), entangledValue G < 1 → |
| ∀ (δ : ℝ), 0 < δ → δ < 1 - entangledValue G → |
| ∀ (k : ℕ), 1 ≤ k → |
| (∀ S : RepeatedStrategy x y a b k, |
| thresholdProbability G δ S ≤ |
| Real.exp (-κ₀ * δ ^ 13 / (1 + Real.log (((a + 1) * (b + 1) : ℕ) : ℝ)) * k)) ∧ |
| thresholdValue (k := k) G δ ≤ |
| Real.exp (-κ₀ * δ ^ 13 / (1 + Real.log (((a + 1) * (b + 1) : ℕ) : ℝ)) * k) := by |
| refine ⟨((2:ℝ)^101)⁻¹, by positivity, ?_⟩ |
| intro x y a b G _ δ hδ hδv k hk |
| have : Nonempty (Fin k) := ⟨⟨0, by omega⟩⟩ |
| have hs : entangledValue G+δ ≤ 1 := by linarith |
| have hstrategy : ∀ S : RepeatedStrategy x y a b k, |
| thresholdProbability G δ S ≤ |
| Real.exp (-((2:ℝ)^101)⁻¹ * δ^13 / (1+Real.log (((a+1)*(b+1) : ℕ) : ℝ))*k) := by |
| intro S |
| rw [S.threshold_eq_model G (by omega) δ] |
| have hh := (S.repetitionModel G).threshold_exp_bound G rfl rfl hδ hs |
| simp only [Fintype.card_fin] at hh |
| refine hh.trans_eq ?_ |
| congr 1 |
| simp only [div_eq_mul_inv, mul_inv_rev] |
| ring |
| refine ⟨hstrategy, ?_⟩ |
| unfold thresholdValue |
| apply csSup_le |
| · let S : RepeatedStrategy x y a b k := Strategy.deterministic (fun _ => 0) (fun _ => 0) |
| exact ⟨_, Set.mem_range_self S⟩ |
| · rintro _ ⟨S,rfl⟩ |
| exact hstrategy S |
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| end ThresholdParallelRepetition |
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| end |
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| end OAI |
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