openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /ClosedBoundedCuts.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.Variance | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ClosedCutProbability | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {E : Type*} | |
| lemma frequently_closedBlock_ae_of_card_le (p : unitInterval) (hp : (p:ℝ)<1) | |
| (S : ℕ → Set E) (hS : ∀ n, (S n).Finite) | |
| (hdis : Pairwise (fun n m => Disjoint (S n) (S m))) (N : ℕ) | |
| (hN : ∀ n, (S n).ncard ≤ N) : | |
| ∀ᵐ ω ∂law (fun _ : E => p), ∃ᶠ n in atTop, ω ∈ closedBlock (S n) := by | |
| let A := fun n => closedBlock (S n) | |
| have hm' : ∀ n, MeasurableSet[blockSigma (S n)] (A n) := | |
| fun n => measurableSet_closedBlock (S n) (hS n).countable | |
| have hm : ∀ n, MeasurableSet (A n) := fun n => blockSigma_le (S n) _ (hm' n) | |
| have hi := independent_block_events (fun _ : E => p) S hdis A hm' | |
| have hprob : ∀ n, (bitMeasure p {False})^N ≤ law (fun _ : E => p) (A n) := by | |
| intro n | |
| rw [law_closedBlock_const p (S n) (hS n)] | |
| exact pow_le_pow_of_le_one zero_le (prob_le_one) (hN n) | |
| have hsum : (∑' n, law (fun _ : E => p) (A n)) = ∞ := by | |
| apply top_unique | |
| calc | |
| ∞ = ∑' _n : ℕ, (bitMeasure p {False})^N := | |
| (ENNReal.tsum_const_eq_top_of_ne_zero | |
| (pow_ne_zero _ (bitMeasure_false_ne_zero p hp))).symm | |
| _ ≤ _ := ENNReal.tsum_le_tsum hprob | |
| have hlim := measure_limsup_eq_one hm hi hsum | |
| have hh : ∀ᵐ ω ∂law (fun _ : E => p), ω ∈ limsup A atTop := | |
| (mem_ae_iff_prob_eq_one (MeasurableSet.measurableSet_limsup hm)).mpr hlim | |
| exact hh.mono (fun _ h => Filter.mem_limsup_iff_frequently_mem.mp h) | |
| end CriticalPercolation.BernoulliProduct | |
| end | |
| end | |
| end OAI | |