openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /TrialProbability.lean
| 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 | |