| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! |
| # Independent-copy amplification for the seed passes |
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|
| The tail estimate is the direct finite-subset argument from Section 8. |
| Independence is used only between copies at a fixed state. The subsequent |
| union bound over states makes no independence assumption between states. |
| -/ |
|
|
| namespace EditApproximation |
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|
| open Finset MeasureTheory ProbabilityTheory |
| open scoped Classical |
|
|
| theorem independent_failure_subset_bound {Ω ι : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (bad : ι → Set Ω) |
| (hindep : iIndepSet bad μ) (copies : Finset ι) (q : ℝ) |
| (hbad : ∀ i ∈ copies, μ.real (bad i) ≤ q) : |
| μ.real (⋂ i ∈ copies, bad i) ≤ q ^ copies.card := by |
| rw [measureReal_def, hindep.meas_biInter, ENNReal.toReal_prod] |
| have hprod := Finset.prod_le_prod₀ (fun i _ => measureReal_nonneg) hbad |
| simpa only [prod_const, measureReal_def] using hprod |
|
|
| theorem independent_failure_count_bound {Ω ι : Type*} [MeasurableSpace Ω] |
| [Fintype ι] [DecidableEq ι] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (bad : ι → Set Ω) |
| (hindep : iIndepSet bad μ) (k : ℕ) (q : ℝ) |
| (hbad : ∀ i, μ.real (bad i) ≤ q) : |
| μ.real {ω | k ≤ (univ.filter fun i => ω ∈ bad i).card} ≤ |
| (Fintype.card ι).choose k * q ^ k := by |
| classical |
| let subsets := (univ : Finset ι).powersetCard k |
| have hcover : {ω | k ≤ (univ.filter fun i => ω ∈ bad i).card} ⊆ |
| ⋃ s ∈ subsets, ⋂ i ∈ s, bad i := by |
| intro ω hω |
| obtain ⟨s, hsub, hcard⟩ := exists_subset_card_eq hω |
| have hs : s ∈ subsets := mem_powersetCard.mpr |
| ⟨fun i hi => mem_univ i, hcard⟩ |
| apply Set.mem_iUnion.mpr |
| refine ⟨s, Set.mem_iUnion.mpr ⟨hs, ?_⟩⟩ |
| exact Set.mem_iInter.mpr fun i => Set.mem_iInter.mpr fun hi => (mem_filter.mp (hsub hi)).2 |
| apply (measureReal_mono (μ := μ) hcover (measure_ne_top μ _)).trans |
| apply (measureReal_biUnion_finset_le subsets _).trans |
| have hbound (s : Finset ι) (hs : s ∈ subsets) : |
| μ.real (⋂ i ∈ s, bad i) ≤ q ^ k := by |
| have hcard := (mem_powersetCard.mp hs).2 |
| simpa only [hcard] using independent_failure_subset_bound μ bad hindep s q (fun i _ => hbad i) |
| have hsum := sum_le_sum hbound |
| simpa only [sum_const, nsmul_eq_mul, subsets, card_powersetCard, card_univ] using hsum |
|
|
| theorem independent_failure_count_binary_bound {Ω ι : Type*} [MeasurableSpace Ω] |
| [Fintype ι] [DecidableEq ι] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (bad : ι → Set Ω) |
| (hindep : iIndepSet bad μ) (k : ℕ) (q : ℝ) (hq : 0 ≤ q) |
| (hbad : ∀ i, μ.real (bad i) ≤ q) : |
| μ.real {ω | k ≤ (univ.filter fun i => ω ∈ bad i).card} ≤ |
| (2 : ℝ) ^ Fintype.card ι * q ^ k := by |
| apply (independent_failure_count_bound μ bad hindep k q hbad).trans |
| apply mul_le_mul_of_nonneg_right _ (pow_nonneg hq k) |
| exact_mod_cast Nat.choose_le_two_pow (Fintype.card ι) k |
|
|
| end EditApproximation |
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|
| end OAI |
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|
