| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.Sectors |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section EvaluationTable |
| variable {D H Q T : Type*} [Group D] [Group H] [Group Q] [Group T] |
|
|
| noncomputable def evaluationModel (q : H →* Q) (e : D →* H) |
| (v : D →* T) (ρ : T →* Q) (hρ : Function.Injective ρ) |
| (he : q.comp e = ρ.comp v) : e.range →* T := |
| e.rangeRestrict.liftOfSurjective e.rangeRestrict_surjective |
| ⟨v,by |
| intro w hw |
| apply hρ |
| have hh := DFunLike.congr_fun he w |
| change q (e w) = ρ (v w) at hh |
| have hw' : e w = 1 := congrArg Subtype.val hw |
| rw [hw',map_one] at hh |
| simpa only [map_one] using hh.symm⟩ |
|
|
| theorem evaluationModel_comp (q : H →* Q) (e : D →* H) |
| (v : D →* T) (ρ : T →* Q) (hρ : Function.Injective ρ) |
| (he : q.comp e = ρ.comp v) : |
| (evaluationModel q e v ρ hρ he).comp e.rangeRestrict = v := |
| e.rangeRestrict.liftOfRightInverse_comp _ _ _ |
|
|
| theorem evaluationModel_projection (q : H →* Q) (e : D →* H) |
| (v : D →* T) (ρ : T →* Q) (hρ : Function.Injective ρ) |
| (he : q.comp e = ρ.comp v) : |
| q.comp e.range.subtype = ρ.comp (evaluationModel q e v ρ hρ he) := by |
| apply MonoidHom.ext |
| intro x |
| obtain ⟨w,rfl⟩ := e.rangeRestrict_surjective x |
| have hh := DFunLike.congr_fun (evaluationModel_comp q e v ρ hρ he) w |
| change evaluationModel q e v ρ hρ he (e.rangeRestrict w) = v w at hh |
| simp only [MonoidHom.comp_apply,hh] |
| exact DFunLike.congr_fun he w |
|
|
| noncomputable def evaluationTable {Ω E : Type*} [Group E] |
| (q : H →* Q) (e : D →* H) (v : D →* (Ω → E)) |
| (ρ : (Ω → E) →* Q) (hρ : Function.Injective ρ) |
| (he : q.comp e = ρ.comp v) (hv : Function.Surjective v) |
| (h : CentralOn q e.range) : ActualLawfulTable q ρ where |
| carrier := e.range |
| law := h |
| model := evaluationModel q e v ρ hρ he |
| onto := by |
| intro s |
| obtain ⟨w,rfl⟩ := hv s |
| exact ⟨e.rangeRestrict w,DFunLike.congr_fun (evaluationModel_comp q e v ρ hρ he) w⟩ |
| projection := evaluationModel_projection q e v ρ hρ he |
|
|
| end EvaluationTable |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
|
|
