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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E Λ : Type*} (G : BondGraph V E) [Group Λ] [MulAction Λ V]
variable (α : Λ → G.Aut) (hα : ∀ g x, (α g).vertex x=g • x)
include α hα in
lemma action_orbit_dense (hc : G.fullGraph.Connected) (S : Set V) (hS : S.Finite)
(hcover : ∀ x, ∃ s∈S, ∃ g : Λ, g • s=x) (o : V) :
∃ c : ℕ, ∀ y, ∃ g : Λ, g • o∈G.vertexBall y c := by
obtain ⟨c,hc'⟩ := G.finite_subset_vertexBall hc o hS
refine ⟨c,?_⟩
intro y
obtain ⟨s,hs,g,rfl⟩ := hcover y
refine ⟨g,?_⟩
rw [←hα g o,←hα g s,Aut.vertexBall_mem]
exact G.vertexBall_comm.mp (hc' hs)
end CriticalPercolation.BondGraph
end
end
end
end OAI
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