openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /PolygonGroupoidTracks.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonGroupoid | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonGroupoidTracks | |
| open Classical CategoryTheory | |
| namespace PolygonObject | |
| variable {a : ℕ} | |
| noncomputable def standard (a m : ℕ) : PolygonObject a := ⟨m,fun _ => ⊤⟩ | |
| noncomputable def standardPointEquiv (a m : ℕ) : (standard a m).Point ≃ TrackPoint a m where | |
| toFun := Subtype.val | |
| invFun x := ⟨x,by trivial⟩ | |
| left_inv _ := rfl | |
| right_inv _ := rfl | |
| noncomputable def fullGroupArrow {m : ℕ} (g : polygonFullGroup a m) : standard a m ⟶ standard a m where | |
| toEquiv := (standardPointEquiv a m).trans (g.val.trans (standardPointEquiv a m).symm) | |
| hasTable := by | |
| obtain ⟨s,hs,hcover⟩ := g.property | |
| let conv : TableChart a m → Chart (a:=a) m m := fun c => ⟨c.source,c.target,c.shift,c.domain⟩ | |
| refine ⟨s.image conv,?_,?_⟩ | |
| · intro c hc | |
| obtain ⟨d,hd,rfl⟩ := Finset.mem_image.mp hc | |
| intro x hx | |
| refine ⟨by trivial,?_⟩ | |
| exact hs d hd x hx | |
| · intro x | |
| obtain ⟨c,hc,hx⟩ := hcover x.val | |
| exact ⟨conv c,Finset.mem_image.mpr ⟨c,hc,rfl⟩,hx⟩ | |
| noncomputable def fullGroupOfArrow {m : ℕ} (g : standard a m ⟶ standard a m) : polygonFullGroup a m := | |
| ⟨(standardPointEquiv a m).symm.trans (g.toEquiv.trans (standardPointEquiv a m)),by | |
| obtain ⟨s,hs,hcover⟩ := g.hasTable | |
| let conv : Chart (a:=a) m m → TableChart a m := fun c => ⟨c.source,c.target,c.shift,c.domain⟩ | |
| refine ⟨s.image conv,?_,?_⟩ | |
| · intro c hc | |
| obtain ⟨d,hd,rfl⟩ := Finset.mem_image.mp hc | |
| intro x hx | |
| obtain ⟨hu,he⟩ := hs d hd x hx | |
| exact he | |
| · intro x | |
| obtain ⟨c,hc,hx⟩ := hcover ((standardPointEquiv _ _).symm x) | |
| exact ⟨conv c,Finset.mem_image.mpr ⟨c,hc,rfl⟩,hx⟩⟩ | |
| @[simp] theorem arrow_fullGroup {m : ℕ} (g : standard a m ⟶ standard a m) : | |
| fullGroupArrow (fullGroupOfArrow g)=g := by | |
| apply Arrow.ext | |
| apply Equiv.ext | |
| intro x | |
| rfl | |
| @[simp] theorem fullGroup_arrow {m : ℕ} (g : polygonFullGroup a m) : | |
| fullGroupOfArrow (fullGroupArrow g)=g := by | |
| apply Subtype.ext | |
| apply Equiv.ext | |
| intro x | |
| rfl | |
| theorem fullGroupArrow_injective {m : ℕ} : Function.Injective (@fullGroupArrow a m) := by | |
| intro g h he | |
| simpa using congrArg fullGroupOfArrow he | |
| theorem fullGroupOfArrow_injective {m : ℕ} : Function.Injective (@fullGroupOfArrow a m) := by | |
| intro g h he | |
| simpa using congrArg fullGroupArrow he | |
| noncomputable def standardAutEquiv (a m : ℕ) : Aut (standard a m) ≃* polygonFullGroup a m where | |
| toFun g := fullGroupOfArrow g.hom | |
| invFun g := asIso (fullGroupArrow g) | |
| left_inv g := by apply Iso.ext; exact arrow_fullGroup g.hom | |
| right_inv g := fullGroup_arrow g | |
| map_mul' g h := by | |
| apply Subtype.ext | |
| apply Equiv.ext | |
| intro x | |
| rfl | |
| end PolygonObject | |
| end PolygonGroupoidTracks | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |