File size: 2,778 Bytes
f205175 | 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 69 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.FreshTrial
import OAI.Probability.CriticalPercolation.Criticality.BlockIndependence
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E V : Type*}
theorem disjoint_trials_no_success [Countable E] (q : E → unitInterval) (F : V → Set E)
(U : Finset V) (hdis : (U : Set V).Pairwise (fun v w => Disjoint (F v) (F w))) :
(law q).real (⋂ v∈U, (allOpen (F v))ᶜ)=
∏ v∈U, (1-(law q).real (allOpen (F v))) := by
have hpair : Pairwise (fun v w : U => Disjoint (F v) (F w)) := by
intro v w hvw
exact hdis v.2 w.2 (fun h => hvw (Subtype.ext h))
have he : (⋂ v∈U, (allOpen (F v))ᶜ)=⋂ v : U, (allOpen (F v))ᶜ := by
ext ω
simp
have hi := (independent_block_family q (fun v : U => F v) hpair).meas_iInter
(fun v => (measurable_allOpen_block (F v)).compl)
rw [he,measureReal_def,hi,ENNReal.toReal_prod]
change (∏ v : U, (law q).real ((allOpen (F v))ᶜ))=_
rw [Finset.prod_coe_sort U (fun v => (law q).real ((allOpen (F v))ᶜ))]
apply Finset.prod_congr rfl
intro v _hv
exact probReal_compl_eq_one_sub ((blockSigma_le _) _ (measurable_allOpen_block (F v)))
theorem disjoint_trials_no_success_le [Countable E] (q : E → unitInterval) (F : V → Set E)
(U : Finset V) (hdis : (U : Set V).Pairwise (fun v w => Disjoint (F v) (F w)))
(K : ℝ) (hK : K ≤ ∑ v∈U, (law q).real (allOpen (F v))) :
(law q).real (⋂ v∈U, (allOpen (F v))ᶜ) ≤ Real.exp (-K) := by
rw [disjoint_trials_no_success q F U hdis]
calc
_ ≤ ∏ v∈U, Real.exp (-(law q).real (allOpen (F v))) := by
apply Finset.prod_le_prod₀
· intro v _hv
exact sub_nonneg.mpr measureReal_le_one
· intro v _hv
simpa only [sub_eq_add_neg,add_comm] using
Real.add_one_le_exp (-(law q).real (allOpen (F v)))
_ = Real.exp (-(∑ v∈U, (law q).real (allOpen (F v)))) := by
rw [← Real.exp_sum,Finset.sum_neg_distrib]
_ ≤ _ := Real.exp_le_exp.mpr (neg_le_neg hK)
lemma bad_trial_union_bound (q : E → unitInterval) (U : Finset V)
(Bad : V → Set (Set E)) (s : V → ℝ) (K a : ℝ) (ha : 0 ≤ a)
(hb : ∀ v∈U, (law q).real (Bad v) ≤ s v*a)
(hs : (∑ v∈U, s v) ≤ K+1) :
(law q).real (⋃ v∈U, Bad v) ≤ (K+1)*a := by
calc
_ ≤ ∑ v∈U, (law q).real (Bad v) := measureReal_biUnion_finset_le U Bad
_ ≤ ∑ v∈U, s v*a := Finset.sum_le_sum hb
_ = (∑ v∈U, s v)*a := by rw [Finset.sum_mul]
_ ≤ _ := mul_le_mul_of_nonneg_right hs ha
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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