openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /FormalSectorProjection.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PairControl | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.FormalSectorSubfamily | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.FormalStarTables | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section FormalSectorProjection | |
| variable {α ι Ω H Q : Type*} [Fintype α] [DecidableEq α] [Group H] [Group Q] | |
| theorem alternating_function_center (hα : 4 ≤ Nat.card α) : | |
| Subgroup.center (Ω → alternatingGroup α)=⊥ := by | |
| rw [Subgroup.center_pi] | |
| simp only [alternatingGroup.center_eq_bot hα,Subgroup.pi_bot] | |
| theorem assignmentPullback_surjective {X Y E : Type*} [Group E] (v : X → Y) | |
| (hv : Function.Injective v) : Function.Surjective (assignmentPullback (E := E) v) := by | |
| classical | |
| intro f | |
| refine ⟨Function.extend v f (fun _ => 1),?_⟩ | |
| ext x | |
| exact congrFun (Function.extend_comp hv f (fun _ => 1)) x | |
| namespace FormalStarTable | |
| variable [Group.IsPerfect (alternatingGroup α)] [Finite ι] | |
| {F : Option ι → TrackStar α →* H} (T : FormalStarTable F) | |
| theorem sector_projection (hα : 4 ≤ Nat.card α) | |
| (q : H →* Q) (ρ : (Ω → alternatingGroup α) →* Q) (hρ : Function.Injective ρ) | |
| (δ : ((ι → Bool) → alternatingGroup α) →* (Ω → alternatingGroup α)) | |
| (hδ : Function.Surjective δ) | |
| (hp : ∀ i, q.comp (F i)=ρ.comp (δ.comp | |
| ((maskFamily (fun j : ι => {σ : ι → Bool | σ j=true}) i).comp (universalProjection _)))) | |
| (V : Set (ι → Bool)) (s : TrackStar α) : | |
| q (T.sector V s)=ρ (δ (sectorMask V (universalProjection _ s))) := by | |
| let e := copyFamilyEval T.input | |
| let v := δ.comp (formalAssignmentEval (universalProjection (alternatingGroup α))) | |
| have hv : Function.Surjective v := hδ.comp (formalAssignmentEval_surjective | |
| (universalProjection (alternatingGroup α)) (universalProjection_surjective _)) | |
| have he : (q.comp T.carrier.subtype).comp e=ρ.comp v := by | |
| rw [MonoidHom.comp_assoc,T.input_eval] | |
| apply FreeGroup.ext_hom | |
| rintro ⟨i,t⟩ | |
| have hh := DFunLike.congr_fun (hp i) t | |
| simpa only [MonoidHom.comp_apply,v,formalAssignmentEval,copySourceMap,FreeGroup.map.of, | |
| assignmentEval,copyFamilyEval_of] using hh | |
| have hk : ∀ w, e w=1 → v w=1 := by | |
| intro w hw | |
| apply hρ | |
| have hh := DFunLike.congr_fun he w | |
| change q ((e w).val)=ρ (v w) at hh | |
| rw [hw,Subgroup.coe_one,map_one] at hh | |
| simpa only [map_one] using hh.symm | |
| let γ : T.carrier →* (Ω → alternatingGroup α) := | |
| e.liftOfSurjective T.input_surjective ⟨v,hk⟩ | |
| have hγe : γ.comp e=v := e.liftOfRightInverse_comp _ _ _ | |
| have hγ : Function.Surjective γ := by | |
| intro x | |
| obtain ⟨w,rfl⟩ := hv x | |
| exact ⟨e w,DFunLike.congr_fun hγe w⟩ | |
| have hγp : q.comp T.carrier.subtype=ρ.comp γ := by | |
| apply (MonoidHom.cancel_right T.input_surjective).mp | |
| change (q.comp T.carrier.subtype).comp e=(ρ.comp γ).comp e | |
| change (q.comp T.carrier.subtype).comp e=ρ.comp (γ.comp e) | |
| rw [hγe] | |
| exact he | |
| have hγi (i : Option ι) : γ.comp (T.input i)=δ.comp | |
| ((maskFamily (fun j : ι => {σ : ι → Bool | σ j=true}) i).comp (universalProjection _)) := by | |
| apply MonoidHom.ext | |
| intro t | |
| have hh := DFunLike.congr_fun hγe (FreeGroup.of (i,t)) | |
| simpa only [MonoidHom.comp_apply,e,v,copyFamilyEval_of,formalAssignmentEval, | |
| copySourceMap,FreeGroup.map.of,assignmentEval] using hh | |
| have hs := T.sector_forward γ hγ (alternating_function_center hα) δ hγi V s | |
| have hh := DFunLike.congr_fun hγp (centralSector T.model V s) | |
| simpa only [MonoidHom.comp_apply,sector,hs] using hh | |
| end FormalStarTable | |
| end FormalSectorProjection | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |