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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 70 71 72 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.TrialProbability
import OAI.Probability.CriticalPercolation.Gluing.SelectedFresh
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E V : Type*} [Countable E] [Countable V]
lemma selected_no_success_bound (q : E → unitInterval) (X Y : Set E)
(hXY : Disjoint X Y) (d : Set E → Finset V) (F : V → Set E)
(hd : ∀ U, MeasurableSet[blockSigma X] {ω | d ω=U})
(hF : ∀ v, F v ⊆ Y) (C : Set (Set E)) (hC : MeasurableSet[blockSigma X] C)
(K : ℝ)
(hdis : ∀ σ∈C, (d σ : Set V).Pairwise (fun v w => Disjoint (F v) (F w)))
(hs : ∀ σ∈C, K ≤ ∑ v∈d σ, (law q).real (allOpen (F v))) :
(law q).real (C∩{ω | ∀ v∈d ω, ω∉allOpen (F v)}) ≤ Real.exp (-K)*(law q).real C := by
have hA (U : Finset V) : MeasurableSet[blockSigma Y] (⋂ v∈U, (allOpen (F v))ᶜ) :=
MeasurableSet.biInter U.countable_toSet (fun v _ =>
((blockSigma_mono (hF v)) _ (measurable_allOpen_block (F v))).compl)
have hh := selected_fresh_bound q X Y hXY d (fun U => ⋂ v∈U, (allOpen (F v))ᶜ)
hd hA C hC (Real.exp (-K)) (Real.exp_nonneg _)
(fun σ hσ => disjoint_trials_no_success_le q F (d σ) (hdis σ hσ) K (hs σ hσ))
simpa only [selectedEvent,Set.mem_iInter,Set.mem_compl_iff] using hh
lemma selected_bad_trial_bound (q : E → unitInterval) (X Y : Set E)
(hXY : Disjoint X Y) (d : Set E → Finset V) (F : V → Set E)
(A : V → Set (Set E)) (hA : ∀ v, MeasurableSet (A v))
(hAu : ∀ v, IsUpperSet (A v))
(hd : ∀ U, MeasurableSet[blockSigma X] {ω | d ω=U})
(hF : ∀ v, F v ⊆ Y) (C : Set (Set E)) (hC : MeasurableSet[blockSigma X] C)
(K η lam : ℝ) (hK : 0 ≤ K+1) (hη : 0 ≤ η) (hlam : 0 < lam)
(hmean : ∀ σ∈C, ∀ v∈d σ, 1-η ≤ (law q).real (A v))
(hs : ∀ σ∈C, (∑ v∈d σ, (law q).real (allOpen (F v))) ≤ K+1) :
(law q).real (C∩{ω | ∃ v∈d ω, ω∈allOpen (F v) ∧
(blockProbability q Y (A v) ω).toReal<1-lam}) ≤
((K+1)*η/lam)*(law q).real C := by
let Bad := fun v => allOpen (F v)∩{σ | (blockProbability q Y (A v) σ).toReal<1-lam}
have hmB (v : V) : MeasurableSet[blockSigma Y] (Bad v) :=
((blockSigma_mono (hF v)) _ (measurable_allOpen_block (F v))).inter
(measurableSet_lt (measurable_blockProbability q Y (A v) (hA v)).ennreal_toReal
measurable_const)
have hb (σ : Set E) (hσ : σ∈C) (v : V) (hv : v∈d σ) : (law q).real (Bad v) ≤ (law q).real (allOpen (F v))*(η/lam) := by
simpa only [mul_div_assoc] using good_trial_bound q Y (F v) (A v) (hA v) (hAu v)
η lam hlam (hmean σ hσ v hv)
have hh := selected_fresh_bound q X Y hXY d (fun U => ⋃ v∈U, Bad v) hd
(fun U => MeasurableSet.biUnion U.countable_toSet (fun v _ => hmB v))
C hC ((K+1)*(η/lam)) (mul_nonneg hK (div_nonneg hη hlam.le))
(fun σ hσ => bad_trial_union_bound q (d σ) Bad (fun v => (law q).real (allOpen (F v)))
K (η/lam) (div_nonneg hη hlam.le) (fun v hv => hb σ hσ v hv) (hs σ hσ))
have he : selectedEvent d (fun U => ⋃ v∈U, Bad v)=
{ω | ∃ v∈d ω, ω∈allOpen (F v) ∧
(blockProbability q Y (A v) ω).toReal<1-lam} := by
ext ω
simp only [selectedEvent, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_ofPred_eq, Bad]
aesop
rw [he] at hh
simpa only [mul_div_assoc] using hh
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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