| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MeasurableSetRestrict |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneIndicatorUpper |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BernoulliProduct |
| variable {E I : Type*} |
|
|
| lemma upper_biUnion (A : I → Set (Set E)) (s : Finset I) |
| (hA : ∀ i∈s, IsUpperSet (A i)) : IsUpperSet (⋃ i∈s, A i) := by |
| intro ω τ h hω |
| obtain ⟨i,hi,hω⟩ := Set.mem_iUnion₂.mp hω |
| exact Set.mem_iUnion₂.mpr ⟨i,hi,hA i hi h hω⟩ |
|
|
| lemma finiteWitness_biUnion (A : I → Set (Set E)) (s : Finset I) |
| (hA : ∀ i∈s, HasFiniteOpenWitness (A i)) : |
| HasFiniteOpenWitness (⋃ i∈s, A i) := by |
| intro ω hω |
| obtain ⟨i,hi,hω⟩ := Set.mem_iUnion₂.mp hω |
| obtain ⟨K,hK,hKs,hKA⟩ := hA i hi ω hω |
| exact ⟨K,hK,hKs,Set.mem_iUnion₂.mpr ⟨i,hi,hKA⟩⟩ |
|
|
| theorem finite_miss_product [Countable E] (q : E → unitInterval) (A : I → Set (Set E)) |
| (s : Finset I) (hA : ∀ i∈s, IsUpperSet (A i)) |
| (hw : ∀ i∈s, HasFiniteOpenWitness (A i)) : |
| ∏ i∈s, (1-(law q).real (A i)) ≤ 1-(law q).real (⋃ i∈s, A i) := by |
| induction s using Finset.induction_on with |
| | empty => simp |
| | @insert i s hi ih => |
| have hm := hA i (Finset.mem_insert_self i s) |
| have hw' := hw i (Finset.mem_insert_self i s) |
| have hs : ∀ j∈s, IsUpperSet (A j) := fun j hj => hA j (Finset.mem_insert_of_mem hj) |
| have hsw : ∀ j∈s, HasFiniteOpenWitness (A j) := fun j hj => hw j (Finset.mem_insert_of_mem hj) |
| have hind := ih hs hsw |
| let B := ⋃ j∈s, A j |
| have hBm := upper_biUnion A s hs |
| have hBw := finiteWitness_biUnion A s hsw |
| have hH := harris_finiteWitness q (A i) B hm hBm hw' hBw |
| have hu := measureReal_union_add_inter (s := A i) |
| (measurableSet_finiteWitness B hBm hBw) (μ := law q) |
| (measure_ne_top _ _) (measure_ne_top _ _) |
| have hb : (law q).real (A i)≤1 := measureReal_le_one |
| have hp := mul_le_mul_of_nonneg_left hind (by linarith : 0≤1-(law q).real (A i)) |
| rw [Finset.prod_insert hi] |
| rw [Finset.set_biUnion_insert] |
| change (1-(law q).real (A i))*(∏ j∈s, (1-(law q).real (A j)))≤ |
| 1-(law q).real (A i∪B) |
| change _≤1-(law q).real B at hind |
| change _≤(1-(law q).real (A i))*(1-(law q).real B) at hp |
| nlinarith |
|
|
| theorem exists_favored_finite [Countable E] (q : E → unitInterval) (A : I → Set (Set E)) |
| (s : Finset I) (hA : ∀ i∈s, IsUpperSet (A i)) |
| (hw : ∀ i∈s, HasFiniteOpenWitness (A i)) (a : ℝ) (ha : 0≤a) |
| (hu : 1-a^s.card<(law q).real (⋃ i∈s, A i)) : |
| ∃ i∈s, 1-a<(law q).real (A i) := by |
| by_contra hn |
| push Not at hn |
| have hp : a^s.card≤∏ i∈s, (1-(law q).real (A i)) := by |
| rw [←Finset.prod_const] |
| apply Finset.prod_le_prod₀ (fun _ _ => ha) |
| intro i hi |
| linarith [hn i hi] |
| have hH := finite_miss_product q A s hA hw |
| linarith |
|
|
| end CriticalPercolation.BernoulliProduct |
| end |
| end |
| end |
|
|
| end OAI |
|
|
