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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Clusters.ClusterObject
import OAI.Probability.CriticalPercolation.GraphGeometry.CutEdges
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.BondGraph
open scoped Classical
variable {V E : Type*}
lemma cutEdges_subset_incidentEdges (G : BondGraph V E) (O : Set V) :
G.cutEdges O ⊆ G.incidentEdges O := by
rintro e ⟨x,y,he,_,hy⟩
exact ⟨y,hy,by simp [he]⟩
lemma eraseVertices_agree_of_cutClosed (G : BondGraph V E) (O : Set V) (ω : Set E)
(hc : ∀ e ∈ G.cutEdges O, ¬ω e) {v z : V} {e : E}
(hv : v ∉ O) (he : G.ends e = s(v,z)) :
ω e = G.eraseVertices ω O e := by
apply propext
constructor
· intro hω
refine ⟨hω,?_⟩
rintro ⟨u,hu,hue⟩
have : u = v ∨ u = z := by simpa only [he,Sym2.mem_iff] using hue
rcases this with rfl | rfl
· exact hv hu
· exact hc e ⟨v,_,he,hv,hu⟩ hω
· exact fun h => h.1
end CriticalPercolation.BondGraph
end
end
end OAI
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