| 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 |
|
|
