File size: 2,998 Bytes
e63eb7f | 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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import Mathlib
namespace OAI
/-!
# Independent-copy amplification for the seed passes
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
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
end OAI
|