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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | import OAI.Probability.EntangledGames.StrategyBridge
namespace OAI
noncomputable section
open scoped BigOperators ComplexOrder
namespace ThresholdParallelRepetition
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
end ThresholdParallelRepetition
end
end OAI
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