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0dd3af4 | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
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