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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 73 74 75 76 77 78 79 80 | 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
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