openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /FiniteKernelOrbits.lean
| 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 | |