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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent
import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy
import OAI.Probability.CriticalPercolation.ProductMeasure.ProductEmbedding
import OAI.Probability.CriticalPercolation.Gluing.SourceGraph
import OAI.Probability.CriticalPercolation.Gluing.SourceLift
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma withSource_reachable_from_seed (A : Set V) (ω : Set (E⊕A))
(hw : ∀ a : A, Sum.inr a∈ω) {v : V}
(hv : ∃ a∈A, (G.openGraph (Sum.inl ⁻¹' ω)).Reachable a v) :
((G.withSource A).openGraph ω).Reachable none (some v) := by
obtain ⟨a,ha,hav⟩ := hv
have hh : ((G.withSource A).openGraph ω).Adj none (some a) :=
⟨⟨Sum.inr ⟨a,ha⟩,hw ⟨a,ha⟩,rfl⟩,by simp⟩
exact hh.reachable.trans (G.withSource_original_reachable A ω hav)
lemma withSource_reachable_to_seed (A : Set V) (ω : Set (E⊕A)) {v : V}
(hv : ((G.withSource A).openGraph ω).Reachable none (some v)) :
∃ a∈A, (G.openGraph (Sum.inl ⁻¹' ω)).Reachable a v := by
have hi : ∀ w, ((G.withSource A).openGraph ω).Reachable none w →
Option.elim' True (fun v => ∃ a∈A, (G.openGraph (Sum.inl ⁻¹' ω)).Reachable a v) w := by
intro w hw
rw [SimpleGraph.reachable_iff_reflTransGen] at hw
induction hw with
| refl => trivial
| @tail u w hr huw ih =>
obtain ⟨⟨b,hb,hend⟩,hne⟩ := huw
cases w with
| none => trivial
| some w =>
rcases G.withSource_edge_cases A b u (some w) hend with
⟨b,v,w',rfl,rfl,hw',hb'⟩|⟨a,rfl,hh⟩
· have he : w'=w := Option.some.inj hw'.symm
subst w'
obtain ⟨a,ha,hav⟩ := ih
refine ⟨a,ha,hav.trans ?_⟩
exact (show (G.openGraph (Sum.inl ⁻¹' ω)).Adj v w from
⟨⟨b,hb,hb'⟩,fun h => hne (congrArg some h)⟩).reachable
· rcases hh with ⟨rfl,hh⟩|⟨hh,_⟩
· have he : w=a.val := Option.some.inj hh
exact ⟨a.val,a.property,he.symm ▸ SimpleGraph.Reachable.refl _⟩
· simp at hh
exact hi (some v) hv
lemma source_hit_iff_seedHit (A T : Set V) (ω : Set (E⊕A))
(hw : ∀ a : A, Sum.inr a∈ω) :
ω∈(G.withSource A).hitEvent none (some '' T) ↔
Sum.inl ⁻¹' ω∈G.seedHitEvent A T := by
constructor
· rintro ⟨w,hr,v,hv,rfl⟩
obtain ⟨a,ha,hav⟩ := G.withSource_reachable_to_seed A ω hr
exact Set.mem_iUnion₂.mpr ⟨a,ha,v,hav,hv⟩
· intro h
obtain ⟨a,ha,v,hav,hv⟩ := Set.mem_iUnion₂.mp h
exact ⟨some v,G.withSource_reachable_from_seed A ω hw ⟨a,ha,hav⟩,v,hv,rfl⟩
lemma source_hit_probability [Countable V] [Countable E] (A T : Set V)
(q : (E⊕A) → unitInterval) (hq : ∀ a : A, q (Sum.inr a)=1) :
BernoulliProduct.law q ((G.withSource A).hitEvent none (some '' T))=
BernoulliProduct.law (fun b => q (Sum.inl b)) (G.seedHitEvent A T) := by
have ha (a : A) : ∀ᵐ ω ∂BernoulliProduct.law q, Sum.inr a∈ω := by
rw [ae_iff]
have he : {ω : Set (E⊕A) | ¬Sum.inr a∈ω}={ω | (Sum.inr a∈ω)=False} := by
ext ω; simp
rw [he,BernoulliProduct.law_coordinate,hq a]
simp [BernoulliProduct.bitMeasure]
have hall : ∀ᵐ ω ∂BernoulliProduct.law q, ∀ a : A, Sum.inr a∈ω := ae_all_iff.mpr ha
have heq : BernoulliProduct.law q ((G.withSource A).hitEvent none (some '' T))=
BernoulliProduct.law q ((fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) ⁻¹' G.seedHitEvent A T) := by
apply measure_congr
filter_upwards [hall] with ω hω
exact propext (G.source_hit_iff_seedHit A T ω hω)
rw [heq]
change BernoulliProduct.law q ((fun ω : Set (E⊕A) => fun b => Sum.inl b∈ω) ⁻¹' G.seedHitEvent A T)=_
rw [← Measure.map_apply
(Measurable.of_eval (fun bond => measurable_set_mem (Sum.inl bond)))
(G.measurableSet_seedHitEvent A T)]
change (Measure.map (fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) (BernoulliProduct.law q)) (G.seedHitEvent A T)=_
rw [BernoulliProduct.law_map_preimage q Sum.inl Sum.inl_injective]
rfl
end CriticalPercolation.BondGraph
end
end
end
end OAI
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