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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.ConditionalGluing
import OAI.Probability.CriticalPercolation.ProductMeasure.ProductEmbedding
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def withSource (A : Set V) : BondGraph (Option V) (E⊕A) where
ends e := match e with
| Sum.inl b => Sym2.map Option.some (G.ends b)
| Sum.inr a => s(none,some a.val)
lemma withSource_local (hl : G.LocallyFinite) {A : Set V} (hA : A.Finite) :
(G.withSource A).LocallyFinite := by
intro v
have hw : (Set.range (Sum.inr : A → E⊕A)).Finite := by
let := hA.fintype
exact Set.finite_range _
cases v with
| none =>
apply hw.subset
intro e he
cases e with
| inl e => simp [withSource] at he
| inr a => exact ⟨a,rfl⟩
| some v =>
apply ((hl v).image Sum.inl |>.union hw).subset
intro e he
cases e with
| inl e =>
apply Or.inl
refine ⟨e,?_,rfl⟩
simpa [withSource] using he
| inr a => exact Or.inr ⟨a,rfl⟩
lemma withSource_original_adj (A : Set V) (ω : Set (E⊕A)) {v w : V}
(h : (G.openGraph (Sum.inl ⁻¹' ω)).Adj v w) :
((G.withSource A).openGraph ω).Adj (some v) (some w) := by
obtain ⟨⟨e,he,hend⟩,hne⟩ := h
exact ⟨⟨Sum.inl e,he,by simp [withSource,hend]⟩,fun h => hne (Option.some.inj h)⟩
lemma withSource_original_reachable (A : Set V) (ω : Set (E⊕A)) {v w : V}
(h : (G.openGraph (Sum.inl ⁻¹' ω)).Reachable v w) :
((G.withSource A).openGraph ω).Reachable (some v) (some w) :=
h.map ⟨Option.some,fun hvw => G.withSource_original_adj A ω hvw⟩
lemma withSource_seedHit [Countable V] [Countable E] (A B T : Set V)
(q : (E⊕A) → unitInterval) :
(BernoulliProduct.law (fun e => q (Sum.inl e))).real (G.seedHitEvent B T) ≤
(BernoulliProduct.law q).real ((G.withSource A).seedHitEvent (Option.some '' B) (Option.some '' T)) := by
have hm : Measurable (fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω : Set (E⊕A) → Set E) :=
Measurable.of_eval (fun edge => measurable_set_mem (Sum.inl edge))
have hlaw : BernoulliProduct.law (fun e => q (Sum.inl e)) =
(BernoulliProduct.law q).map (fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) :=
(BernoulliProduct.law_map_preimage q Sum.inl Sum.inl_injective).symm
rw [hlaw,measureReal_def,
Measure.map_apply hm (G.measurableSet_seedHitEvent B T)]
change (BernoulliProduct.law q).real ((fun ω => Sum.inl ⁻¹' ω) ⁻¹' G.seedHitEvent B T) ≤ _
refine measureReal_mono ?_ (measure_ne_top _ _)
intro ω hω
obtain ⟨v,hv,w,hvw,hw⟩ := Set.mem_iUnion₂.mp hω
exact Set.mem_iUnion₂.mpr ⟨some v,⟨v,hv,rfl⟩,some w,
G.withSource_original_reachable A ω hvw,⟨w,hw,rfl⟩⟩
lemma withSource_incident (A : Set V) {v : V} (hv : v∉A) :
(G.withSource A).incidentEdges {some v}=Sum.inl '' G.incidentEdges {v} := by
ext e
cases e with
| inl e => simp [incidentEdges,withSource]
| inr a =>
have hne : v≠a.val := fun hh => hv (hh.symm ▸ a.property)
simp [incidentEdges,withSource,hne]
lemma withSource_trial_reachable (A : Set V) (F : Set E) {v w : V}
(h : (G.openGraph F).Reachable v w) :
((G.withSource A).openGraph (Sum.inl '' F)).Reachable (some v) (some w) := by
apply G.withSource_original_reachable
simpa only [Set.preimage_image_eq _ Sum.inl_injective] using h
end CriticalPercolation.BondGraph
end
end
end
end OAI
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