| import OAI.Probability.EntangledGames.TestExtraction |
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| namespace OAI |
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| noncomputable section |
| namespace ThresholdParallelRepetition |
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| lemma power_ratio_bound {v δ : ℝ} (hv : 0 ≤ v) (hδ : 0 < δ) (hs : v+δ ≤ 1) (T : ℕ) : |
| (v+δ/2)^T/(v+δ)^T ≤ Real.exp (-(δ*(T : ℝ)/2)) := by |
| have hs0 : 0 < v+δ := by linarith |
| have hr0 : 0 ≤ (v+δ/2)/(v+δ) := div_nonneg (by linarith) hs0.le |
| have hr : (v+δ/2)/(v+δ) ≤ Real.exp (-δ/2) := by |
| have he := Real.add_one_le_exp (-δ/2) |
| have hratio : (v+δ/2)/(v+δ) ≤ 1-δ/2 := by |
| apply (div_le_iff₀ hs0).mpr |
| nlinarith |
| linarith |
| rw [← div_pow] |
| have hh := pow_le_pow_left₀ hr0 hr T |
| rw [← Real.exp_nat_mul] at hh |
| exact hh.trans_eq (by congr 1; ring) |
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| lemma inverse_power_bound {s δ : ℝ} (hδ : 0 < δ) (hs : δ ≤ s) (T : ℕ) : |
| 1/s^T ≤ Real.exp ((T : ℝ)/δ) := by |
| have hs0 : 0 < s := hδ.trans_le hs |
| have hh := Real.log_le_sub_one_of_pos (one_div_pos.mpr hδ) |
| rw [one_div, Real.log_inv] at hh |
| have hlog := Real.log_le_log hδ hs |
| have hl : -Real.log s ≤ 1/δ := by simpa only [one_div] using (show -Real.log s ≤ δ⁻¹ by linarith) |
| have hpow := mul_le_mul_of_nonneg_left hl (Nat.cast_nonneg T : (0:ℝ) ≤ T) |
| have he : 1/s^T = Real.exp (-(T : ℝ)*Real.log s) := by |
| rw [neg_mul, Real.exp_neg, Real.exp_nat_mul, Real.exp_log hs0, one_div] |
| rw [he] |
| apply Real.exp_le_exp.mpr |
| simpa only [mul_neg, neg_mul, div_eq_mul_inv, one_mul] using hpow |
|
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| end ThresholdParallelRepetition |
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| end |
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|
| noncomputable section |
| namespace ThresholdParallelRepetition |
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|
| lemma analytic_threshold_bound {v δ θ R E q : ℝ} |
| (hv : 0 ≤ v) (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hs : v+δ ≤ 1) |
| (hθ : 2 ≤ θ) (hδθ : 8 ≤ δ*θ) |
| (hθR : θ/δ ≤ R/4) (hR : 2/δ ≤ R/4) (hE : E ≤ δ*θ/8) |
| (hq : q*(v+δ)^(⌊θ⌋₊) ≤ (v+δ/2)^(⌊θ⌋₊)+Real.exp (-R)*(2/δ)) : |
| q ≤ Real.exp (-E) := by |
| let T := ⌊θ⌋₊ |
| let L := δ*θ/4 |
| have hθ0 : 0 ≤ θ := by linarith |
| have hT : (T : ℝ) ≤ θ := Nat.floor_le hθ0 |
| have hT' : θ/2 ≤ (T : ℝ) := by |
| have hh := Nat.lt_floor_add_one θ |
| dsimp [T]; linarith |
| have hs0 : 0 < v+δ := by linarith |
| have hp0 : 0 < (v+δ)^T := pow_pos hs0 _ |
| have hq' : q ≤ (v+δ/2)^T/(v+δ)^T + Real.exp (-R)*(2/δ)/(v+δ)^T := by |
| rw [← add_div] |
| exact (le_div_iff₀ hp0).mpr hq |
| have hfirst : (v+δ/2)^T/(v+δ)^T ≤ Real.exp (-L) := by |
| apply (power_ratio_bound hv hδ hs T).trans |
| apply Real.exp_le_exp.mpr |
| dsimp [L] |
| nlinarith |
| have hTδ : (T : ℝ)/δ ≤ R/4 := (div_le_div_of_nonneg_right hT hδ.le).trans hθR |
| have hfac : 2/δ ≤ Real.exp (2/δ) := by linarith [Real.add_one_le_exp (2/δ)] |
| have hbad : Real.exp (-R)*(2/δ)/(v+δ)^T ≤ Real.exp (-R/2) := by |
| calc |
| Real.exp (-R)*(2/δ)/(v+δ)^T = Real.exp (-R)*(2/δ)*(1/(v+δ)^T) := by ring |
| _ ≤ Real.exp (-R)*Real.exp (2/δ)*Real.exp ((T : ℝ)/δ) := by |
| apply mul_le_mul |
| · exact mul_le_mul_of_nonneg_left hfac (Real.exp_pos _).le |
| · exact inverse_power_bound hδ (by linarith) T |
| · positivity |
| · positivity |
| _ = Real.exp (-R+2/δ+(T : ℝ)/δ) := by rw [Real.exp_add, Real.exp_add] |
| _ ≤ Real.exp (-R/2) := Real.exp_le_exp.mpr (by linarith) |
| have hδ2 : δ*δ ≤ 1 := by nlinarith |
| have hL : L ≤ R/2 := by |
| have hh : δ*θ ≤ θ/δ := by |
| apply (le_div_iff₀ hδ).mpr |
| nlinarith [mul_le_mul_of_nonneg_right hδ2 hθ0] |
| have hR0 : 0 < R := by have := div_pos (show (0:ℝ)<2 by norm_num) hδ; linarith |
| dsimp [L]; linarith |
| have hqL : q ≤ 2*Real.exp (-L) := by |
| have hb : Real.exp (-R)*(2/δ)/(v+δ)^T ≤ Real.exp (-L) := |
| hbad.trans (Real.exp_le_exp.mpr (by linarith)) |
| linarith |
| have htwo : 2 ≤ Real.exp (L/2) := by |
| have hh := Real.add_one_le_exp (L/2) |
| dsimp [L] at * |
| linarith |
| calc |
| q ≤ 2*Real.exp (-L) := hqL |
| _ ≤ Real.exp (L/2)*Real.exp (-L) := mul_le_mul_of_nonneg_right htwo (Real.exp_pos _).le |
| _ = Real.exp (-L/2) := by rw [← Real.exp_add]; congr 1; ring |
| _ ≤ Real.exp (-E) := by apply Real.exp_le_exp.mpr; dsimp [L]; linarith |
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| lemma large_rate_parameters {δ k D : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1) |
| (hk : 0 < k) (hD : 1 ≤ D) (hbig : (2:ℝ)^100*D ≤ δ^12*k) : |
| 2 ≤ δ^9*k/((2:ℝ)^90*D) ∧ |
| 8 ≤ δ*(δ^9*k/((2:ℝ)^90*D)) ∧ |
| (δ^9*k/((2:ℝ)^90*D))/δ ≤ (δ^8*k/(2:ℝ)^83)/4 ∧ |
| 2/δ ≤ (δ^8*k/(2:ℝ)^83)/4 ∧ |
| δ^13*k/((2:ℝ)^101*D) ≤ δ*(δ^9*k/((2:ℝ)^90*D))/8 := by |
| have hD0 : 0 < D := by linarith |
| have hden : 0 < (2:ℝ)^90*D := by positivity |
| have h129 : δ^12 ≤ δ^9 := pow_le_pow_of_le_one hδ.le hδ1 (by omega) |
| have h1210 : δ^12 ≤ δ^10 := pow_le_pow_of_le_one hδ.le hδ1 (by omega) |
| have h1310 : δ^13 ≤ δ^10 := pow_le_pow_of_le_one hδ.le hδ1 (by omega) |
| have h129k := mul_le_mul_of_nonneg_right h129 hk.le |
| have h1210k := mul_le_mul_of_nonneg_right h1210 hk.le |
| have h1310k := mul_le_mul_of_nonneg_right h1310 hk.le |
| have h8k : 0 ≤ δ^8*k := by positivity |
| have h10k : 0 ≤ δ^10*k := by positivity |
| refine ⟨?_, ?_, ?_, ?_, ?_⟩ |
| · apply (le_div_iff₀ hden).mpr |
| nlinarith |
| · rw [← mul_div_assoc] |
| apply (le_div_iff₀ hden).mpr |
| nlinarith |
| · apply (div_le_iff₀ hδ).mpr |
| apply (div_le_iff₀ hden).mpr |
| have hh := mul_le_mul_of_nonneg_left hD h8k |
| nlinarith |
| · apply (div_le_iff₀ hδ).mpr |
| nlinarith |
| · rw [← mul_div_assoc, div_div] |
| apply (div_le_div_iff₀ (by positivity : 0 < (2:ℝ)^101*D) (by positivity : 0 < (2:ℝ)^90*D*8)).mpr |
| have hh := mul_le_mul_of_nonneg_right h1310k hD0.le |
| nlinarith [mul_nonneg h10k hD0.le] |
|
|
| end ThresholdParallelRepetition |
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| end |
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|
| noncomputable section |
| open scoped BigOperators MatrixOrder ComplexOrder |
| open Matrix |
| namespace ThresholdParallelRepetition |
| open QuantumSampling FiniteProbability Law MixedExposure |
| namespace RepeatedModel |
| variable {I m n : Type} [Fintype I] [DecidableEq I] [Nonempty I] |
| [Fintype m] [DecidableEq m] [Nonempty m] [Fintype n] [DecidableEq n] [Nonempty n] |
| {x y a b : ℕ} (G : Game x y a b) |
| (M : RepeatedModel (Fin (x+1)) (Fin (y+1)) (Fin (a+1)) (Fin (b+1)) I m n) |
| (hμ : M.μ = G.questionLaw) (hV : M.V = G.accepts) |
| include hμ hV |
|
|
| lemma threshold_exp_bound {δ : ℝ} (hδ : 0 < δ) (hs : entangledValue G+δ ≤ 1) : |
| Tests.threshold M.joint M.winsAt (entangledValue G+δ) ≤ |
| Real.exp (-(δ^13*(Fintype.card I : ℝ)/((2:ℝ)^101*(1+Real.log (((a+1)*(b+1) : ℕ) : ℝ))))) := by |
| let D : ℝ := 1+Real.log (((a+1)*(b+1) : ℕ) : ℝ) |
| let k : ℝ := Fintype.card I |
| let θ : ℝ := δ^9*k/((2:ℝ)^90*D) |
| let R : ℝ := δ^8*k/(2:ℝ)^83 |
| let E : ℝ := δ^13*k/((2:ℝ)^101*D) |
| have hd : (1:ℝ) ≤ (((a+1)*(b+1) : ℕ) : ℝ) := by |
| exact_mod_cast (Nat.mul_pos (Nat.succ_pos a) (Nat.succ_pos b)) |
| have hD : 1 ≤ D := by dsimp [D]; linarith [Real.log_nonneg hd] |
| have hD0 : 0 < D := by linarith |
| have hk : 0 < k := by dsimp [k]; exact_mod_cast Fintype.card_pos (α := I) |
| have hv := entangledValue_nonneg G |
| have hδ1 : δ ≤ 1 := by linarith |
| change _ ≤ Real.exp (-E) |
| by_cases hsmall : δ^12*k ≤ (2:ℝ)^100*D |
| · apply (M.threshold_mean_bound G hμ hV hδ hs).trans |
| apply Real.exp_le_exp.mpr |
| suffices E ≤ δ by linarith |
| dsimp [E] |
| apply (div_le_iff₀ (by positivity : 0 < (2:ℝ)^101*D)).mpr |
| have hh := mul_le_mul_of_nonneg_left hsmall hδ.le |
| nlinarith [mul_nonneg hδ.le hD0.le] |
| · obtain ⟨hθ,hδθ,hθR,hR,hE⟩ := large_rate_parameters hδ hδ1 hk hD (le_of_not_ge hsmall) |
| have hθ0 : 0 ≤ θ := by dsimp [θ]; positivity |
| have htest := M.threshold_power_bound G hμ hV hδ hs ⌊θ⌋₊ (Nat.floor_le hθ0) |
| exact analytic_threshold_bound hv hδ hδ1 hs hθ hδθ hθR hR hE htest |
|
|
| end RepeatedModel |
| end ThresholdParallelRepetition |
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| end |
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| end OAI |
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