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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 | import Mathlib
namespace OAI
section
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.EquivariantCoordinates
variable {Λ N T V : Type*} [Group Λ] [Group N] [Group T] [MulAction Λ V]
theorem finite_orbit_of_finite_image (ρ : Λ →* N) (L : Subgroup Λ)
(hfin : (L.map ρ : Set N).Finite)
(hker : ∀ v : V, (MulAction.orbit ρ.ker v).Finite) (v : V) :
(MulAction.orbit L v).Finite := by
have hex : ∀ a : L.map ρ, ∃ g : Λ, g∈L ∧ ρ g=a := by
intro a
exact a.property
choose r hr using hex
let : Finite (L.map ρ) := hfin.to_subtype
have hset : (⋃ a : L.map ρ, (fun x : V => r a • x) '' MulAction.orbit ρ.ker v).Finite :=
Set.finite_iUnion (fun a => (hker v).image _)
apply hset.subset
rintro w ⟨g,hg⟩
change (g : Λ) • v=w at hg
let a : L.map ρ := ⟨ρ g,⟨g,g.property,rfl⟩⟩
have hk : (r a)⁻¹*(g : Λ)∈ρ.ker := by
change ρ ((r a)⁻¹*(g : Λ))=1
rw [map_mul,map_inv,(hr a).2]
exact inv_mul_cancel (ρ g)
apply Set.mem_iUnion.mpr
refine ⟨a,((r a)⁻¹*(g : Λ)) • v,?_,?_⟩
· exact ⟨⟨(r a)⁻¹*(g : Λ),hk⟩,rfl⟩
· change r a • (((r a)⁻¹*(g : Λ)) • v)=w
rw [← mul_smul,mul_inv_cancel_left,hg]
lemma finite_composite_kernel_orbits (ρ : Λ →* N) (φ : N →* T)
(hφ : (φ.ker : Set N).Finite)
(hker : ∀ v : V, (MulAction.orbit ρ.ker v).Finite) (v : V) :
(MulAction.orbit (φ.comp ρ).ker v).Finite := by
apply finite_orbit_of_finite_image ρ _ _ hker v
apply hφ.subset
rintro n ⟨g,hg,rfl⟩
exact hg
lemma finite_vertices_of_finite_quotient (ρ : Λ →* N) [Finite N]
(hker : ∀ v : V, (MulAction.orbit ρ.ker v).Finite)
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) :
(Set.univ : Set V).Finite := by
have hf : ∀ s : V, (MulAction.orbit Λ s).Finite := by
intro s
have hh := finite_orbit_of_finite_image ρ ⊤ (Set.toFinite _) hker s
apply hh.subset
rintro v ⟨g,hg⟩
exact ⟨⟨g,Subgroup.mem_top g⟩,hg⟩
apply (hS.biUnion fun s _ => hf s).subset
intro v _
obtain ⟨s,hs,hvs⟩ := hcover v
exact Set.mem_iUnion₂.mpr ⟨s,hs,hvs⟩
end CriticalPercolation.EquivariantCoordinates
end
end
end
end OAI
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