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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.SliceExploration
import OAI.Probability.CriticalPercolation.Gluing.StripMoves
import OAI.Probability.CriticalPercolation.ProductMeasure.MeasurableSetRestrict
namespace OAI
section
noncomputable section
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical ENNReal Topology
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def freshExitEvent (S : Set V) (v : V) (N : ℕ) : Set (Set E) :=
{ω | ∃ F : Finset E, N≤F.card ∧
(F : Set E)⊆G.freshClusterBoundary (G.inducedEdges S) v ω∩ω}
lemma freshClusterBoundary_induced_mono {S : Set V} {v : V} (hv : v∈S)
{ω η : Set E} (hωη : ω⊆η) :
G.freshClusterBoundary (G.inducedEdges S) v ω ⊆
G.freshClusterBoundary (G.inducedEdges S) v η := by
rintro e ⟨⟨x,y,he,hx,hy⟩,heS⟩
have hyS : y∈S := G.cluster_induced_subset ω S hv hy
have hxS : x∉S := by
intro hxS
apply heS
intro z hz
rw [he] at hz
rcases Sym2.mem_iff.mp hz with rfl|rfl
· exact hxS
· exact hyS
refine ⟨⟨x,y,he,fun hx' => hxS (G.cluster_induced_subset η S hv hx'),?_⟩,heS⟩
exact G.cluster_mono (Set.inter_subset_inter_left _ hωη) v hy
lemma freshExitEvent_upper {S : Set V} {v : V} (hv : v∈S) (N : ℕ) :
IsUpperSet (G.freshExitEvent S v N) := by
rintro ω η hωη ⟨F,hF,hFω⟩
exact ⟨F,hF,fun e he =>
⟨G.freshClusterBoundary_induced_mono hv hωη (hFω he).1,hωη (hFω he).2⟩⟩
lemma freshExitEvent_finiteWitness {S : Set V} {v : V} (hv : v∈S) (N : ℕ) :
BernoulliProduct.HasFiniteOpenWitness (G.freshExitEvent S v N) := by
rintro ω ⟨F,hF,hFω⟩
let X := G.inducedEdges S
let C := G.cluster (ω∩X) v
let T : Set V := (⋃ e∈(F : Set E), (↑(G.ends e).toFinset : Set V))∩C
have hT : T.Finite :=
(F.finite_toSet.biUnion (fun e _ => (G.ends e).toFinset.finite_toSet)).inter_of_left C
obtain ⟨J,hJ,hJω,hconn⟩ := G.finite_subset_cluster_finite_witness hT Set.inter_subset_right
have hsub : J∪(F : Set E)⊆ω := Set.union_subset
(hJω.trans Set.inter_subset_left) (fun e he => (hFω he).2)
refine ⟨J∪(F : Set E),hJ.union F.finite_toSet,hsub,F,hF,fun e he => ?_⟩
obtain ⟨⟨x,y,hexy,_,hy⟩,heX⟩ := (hFω he).1
have hyT : y∈T := by
refine ⟨Set.mem_iUnion₂.mpr ⟨e,he,?_⟩,hy⟩
have hm : y∈G.ends e := by rw [hexy]; simp
simpa only [Finset.mem_coe,Sym2.mem_toFinset] using hm
have hyJ : y∈G.cluster ((J∪(F : Set E))∩X) v :=
G.cluster_mono (Set.subset_inter Set.subset_union_left
(hJω.trans Set.inter_subset_right)) v (hconn hyT)
have hx' : x∉G.cluster ((J∪(F : Set E))∩X) v := by
intro hreach
apply heX
intro vertex hvertex
rw [hexy] at hvertex
rcases Sym2.mem_iff.mp hvertex with rfl | rfl
· exact G.cluster_induced_subset (J∪(F : Set E)) S hv hreach
· exact G.cluster_induced_subset ω S hv hy
exact ⟨⟨⟨x,y,hexy,hx',hyJ⟩,heX⟩,Or.inr he⟩
lemma measurableSet_freshExitEvent [Countable E] {S : Set V} {v : V}
(hv : v∈S) (N : ℕ) : MeasurableSet (G.freshExitEvent S v N) :=
BernoulliProduct.measurableSet_finiteWitness _ (G.freshExitEvent_upper hv N)
(G.freshExitEvent_finiteWitness hv N)
lemma freshExitEvent_of_card (S : Set V) (v : V) (N : ℕ) (ω : Set E)
(hf : (G.freshClusterBoundary (G.inducedEdges S) v ω∩ω).Finite)
(hN : N≤(G.freshClusterBoundary (G.inducedEdges S) v ω∩ω).ncard) :
ω∈G.freshExitEvent S v N := by
exact ⟨hf.toFinset,by rwa [← Set.ncard_eq_toFinset_card _ hf],
fun e he => hf.mem_toFinset.mp he⟩
end CriticalPercolation.BondGraph
end
end
end OAI
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