File size: 5,718 Bytes
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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.ContinuousExpectWeights
import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent
namespace OAI
section
section
open scoped BigOperators
open scoped Classical ENNReal
open Set
open scoped Classical Topology
open Set MeasureTheory ProbabilityTheory
open scoped Classical
open Filter Topology
open Set MeasureTheory ProbabilityTheory Filter
open scoped Topology Classical
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
lemma continuous_bitWeight (b : Prop) : Continuous (fun p : unitInterval => bitWeight p b) := by
by_cases hb : b
· simpa [bitWeight, hb] using (continuous_subtype_val : Continuous (fun p : unitInterval => (p : ℝ)))
· simp only [bitWeight, ite_eq_right hb]
fun_prop
lemma continuous_real_law_finite [Fintype E] (A : Set (Set E)) :
Continuous (fun p : unitInterval => (law (fun _ : E => p)).real A) := by
have h (p : unitInterval) : (law (fun _ : E => p)).real A =
FiniteProduct.expect (fun _ : E => bitWeight p)
(fun ω : E → Prop => A.indicator (fun _ => (1 : ℝ)) {e | ω e}) := by
rw [← integral_law_eq_expect]
exact (integral_indicator_one (.of_discrete) :
(∫ ω, A.indicator (1 : Set E → ℝ) ω ∂law (fun _ : E => p)) = _).symm
simp_rw [h]
apply (FiniteProduct.continuous_expect_weights _).comp
exact continuous_pi fun _ => continuous_pi fun b => continuous_bitWeight b
lemma real_law_restrict (p : unitInterval) (s : Set E) (hs : s.Finite)
(A : Set (Set E)) :
(law (fun _ : E => p)).real {ω | s ∩ ω ∈ A} =
(law (fun _ : s => p)).real {ω | Subtype.val '' ω ∈ A} := by
let := hs.fintype
have hi (ω : Set E) : Subtype.val '' (Subtype.val ⁻¹' ω : Set s) = s ∩ ω := by
ext e
simp
rw [← law_map_restrict (fun _ : E => p) s]
have hf : Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) :=
Measurable.of_eval (fun edge => measurable_set_mem edge.val)
simp only [Measure.real]
rw [Measure.map_apply hf (.of_discrete)]
simp only [Set.preimage_ofPred_eq, hi]
lemma continuous_real_law_restrict (s : Set E) (hs : s.Finite) (A : Set (Set E)) :
Continuous (fun p : unitInterval => (law (fun _ : E => p)).real
{ω | s ∩ ω ∈ A}) := by
let := hs.fintype
simp_rw [real_law_restrict _ s hs A]
exact continuous_real_law_finite _
def HasFiniteOpenWitness (A : Set (Set E)) : Prop :=
∀ ω ∈ A, ∃ s : Set E, s.Finite ∧ s ⊆ ω ∧ s ∈ A
lemma finiteWitness_exhaustion (A : Set (Set E)) (hmono : IsUpperSet A)
(hw : HasFiniteOpenWitness A) (K : Set.FiniteExhaustion (Set.univ : Set E)) :
(⋃ n, {ω | K n ∩ ω ∈ A}) = A := by
ext ω
constructor
· intro h
obtain ⟨n, hn⟩ := Set.mem_iUnion.mp h
exact hmono Set.inter_subset_right hn
· intro h
obtain ⟨s, hs, hsω, hsA⟩ := hw ω h
obtain ⟨n, hn⟩ := BondGraph.finite_subset_exhaustion K hs (Set.subset_univ _)
exact Set.mem_iUnion.mpr ⟨n, hmono (Set.subset_inter hn hsω) hsA⟩
lemma finiteWitness_exhaustion_mono (A : Set (Set E)) (hmono : IsUpperSet A)
(K : Set.FiniteExhaustion (Set.univ : Set E)) :
Monotone (fun n => {ω | K n ∩ ω ∈ A}) := by
intro m n h ω hω
exact hmono (Set.inter_subset_inter_left _ (K.mono h)) hω
lemma tendsto_real_finiteWitness (p : unitInterval) (A : Set (Set E))
(hmono : IsUpperSet A) (hw : HasFiniteOpenWitness A)
(K : Set.FiniteExhaustion (Set.univ : Set E)) :
Tendsto (fun n => (law (fun _ : E => p)).real {ω | K n ∩ ω ∈ A})
atTop (𝓝 ((law (fun _ : E => p)).real A)) := by
have h := tendsto_measure_iUnion_atTop (μ := law (fun _ : E => p))
(finiteWitness_exhaustion_mono A hmono K)
rw [finiteWitness_exhaustion A hmono hw K] at h
exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
lemma lowerSemicontinuous_real_finiteWitness [Countable E] (A : Set (Set E))
(hmono : IsUpperSet A) (hw : HasFiniteOpenWitness A) :
LowerSemicontinuous (fun p : unitInterval => (law (fun _ : E => p)).real A) := by
let K := (Set.to_countable (Set.univ : Set E)).finiteExhaustion
intro p a ha
have ht := (tendsto_real_finiteWitness p A hmono hw K).eventually (lt_mem_nhds ha)
obtain ⟨n, hn⟩ := ht.exists
have hc := (continuous_real_law_restrict (K n) (K.finite n) A).continuousAt (x := p)
filter_upwards [hc.eventually (lt_mem_nhds hn)] with q hq
exact lt_of_lt_of_le hq (measureReal_mono (fun ω hω => hmono Set.inter_subset_right hω))
lemma exists_lower_parameter [Countable E] {I : Type*} [Finite I]
(A : I → Set (Set E)) (hmono : ∀ i, IsUpperSet (A i))
(hw : ∀ i, HasFiniteOpenWitness (A i)) (p : unitInterval) (hp : 0 < p)
(a : I → ℝ) (ha : ∀ i, a i < (law (fun _ : E => p)).real (A i)) :
∃ q : unitInterval, (p : ℝ) / 2 ≤ q ∧ q < p ∧
∀ i, a i < (law (fun _ : E => q)).real (A i) := by
have hall : ∀ᶠ q in 𝓝 p, ∀ i, a i < (law (fun _ : E => q)).real (A i) :=
Filter.eventually_all.mpr fun i =>
lowerSemicontinuous_real_finiteWitness (A i) (hmono i) (hw i) p (a i) (ha i)
have hp' : (0 : ℝ) < p := hp
have hhalf : ∀ᶠ q : unitInterval in 𝓝 p, (p : ℝ) / 2 < q :=
continuous_subtype_val.continuousAt.eventually (lt_mem_nhds (by linarith : (p : ℝ) / 2 < p))
have : NeBot (𝓝[<] p) := nhdsLT_neBot_of_exists_lt ⟨0, hp⟩
obtain ⟨q, hq, hqp, hqA⟩ :=
((hhalf.filter_mono nhdsWithin_le_nhds).and
((show ∀ᶠ q in 𝓝[<] p, q < p from self_mem_nhdsWithin).and
(hall.filter_mono nhdsWithin_le_nhds))).exists
exact ⟨q, le_of_lt hq, hqp, hqA⟩
end CriticalPercolation.BernoulliProduct
end
end
end OAI
|