File size: 6,047 Bytes
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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.FrameDisjointness
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section DistinctSlotStars
noncomputable def fiveAlternatingHom {m : ℕ} (ι : Fin 5 ↪ Fin m) :
alternatingGroup (Fin 5) →* alternatingGroup (Fin m) :=
(fiveTrackHom ι).codRestrict _ (fiveTrack_sign ι)
noncomputable def orderedTrackAlphabet {m : ℕ} (ι : Fin 5 ↪ Fin m) : Finset (Fin m) :=
Finset.univ.map ι
instance orderedTrackAlphabetPerfect {m : ℕ} (ι : Fin 5 ↪ Fin m) :
Group.IsPerfect (alternatingGroup (orderedTrackAlphabet ι)) := by
classical
apply Group.IsPerfect.mk
apply commutator_alternatingGroup_eq_top
rw [Nat.card_eq_fintype_card,Fintype.card_coe,orderedTrackAlphabet,Finset.card_map,Finset.card_univ,Fintype.card_fin]
noncomputable def orderedTrackHom {m : ℕ} (ι : Fin 5 ↪ Fin m) :
alternatingGroup (Fin 5) →* alternatingGroup (orderedTrackAlphabet ι) :=
(MonoidHom.ofInjective (subtypeAlternatingHom_injective (orderedTrackAlphabet ι))).symm.toMonoidHom.comp
((fiveAlternatingHom ι).codRestrict (subtypeAlternatingHom (orderedTrackAlphabet ι)).range
(fun s => (subtypeAlternatingHom_mem_range _ _).mpr (alphabet_support_subset ι s)))
@[simp] theorem orderedTrackHom_comp {m : ℕ} (ι : Fin 5 ↪ Fin m) :
(subtypeAlternatingHom (orderedTrackAlphabet ι)).comp (orderedTrackHom ι) =
fiveAlternatingHom ι := by
ext s : 1
exact MonoidHom.apply_ofInjective_symm (subtypeAlternatingHom_injective _) _
@[simp] theorem orderedTrackAlphabet_mem {m : ℕ} (ι : Fin 5 ↪ Fin m) (j : Fin 5) :
ι j ∈ orderedTrackAlphabet ι := Finset.mem_map.mpr ⟨j,Finset.mem_univ _,rfl⟩
theorem orderedTrackHom_apply {m : ℕ} (ι : Fin 5 ↪ Fin m)
(s : alternatingGroup (Fin 5)) (j : Fin 5) :
((orderedTrackHom ι s).val ⟨ι j,orderedTrackAlphabet_mem ι j⟩).val = ι (s.val j) := by
have he := congrArg (fun t : alternatingGroup (Fin m) => t.val (ι j))
(DFunLike.congr_fun (orderedTrackHom_comp ι) s)
change Equiv.Perm.ofSubtype (orderedTrackHom ι s).val (ι j) =
Equiv.Perm.viaEmbedding s.val ι (ι j) at he
rw [Equiv.Perm.ofSubtype_apply_of_mem _ (orderedTrackAlphabet_mem ι j),
Equiv.Perm.viaEmbedding_apply] at he
exact he
noncomputable def slotAlternatingHom {a m : ℕ} (V : polygonAlgebra a)
(slots : Fin 5 → Fin m × (CutRing × CutRing))
(hinj : Function.Injective (SlotMap a m V slots)) :
alternatingGroup (Fin 5) →* polygonAlternatingGroup a m :=
((slotHom V slots hinj).comp (alternatingGroup (Fin 5)).subtype).codRestrict _
(fun s => slotHom_alternating_mem V slots hinj s.property)
namespace InitialCoverSystem
variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m}
(B : InitialCoverSystem a r m hm M)
[Group.IsPerfect (alternatingGroup (Fin (m+1)))]
(hlarge : 15 < m+1) (h : B.AllPrimitiveLaws) (hr : 0<ordinary r ∧ ordinary r<1/2)
noncomputable def distinctSlotStar (ι : Fin 5 ↪ Fin (m+1))
(u : Fin (m+1) → CutRing × CutRing)
(F : OffsetFrame a r m hm (orderedTrackAlphabet ι) u) (V : polygonAlgebra a) :
UniversalExtension (alternatingGroup (Fin 5)) →* BoundedRelationCover M (alternatingGenerator a r m hm) :=
(B.frameStar hlarge h hr (orderedTrackAlphabet ι) u F V).comp (universalMap (orderedTrackHom ι))
theorem distinctSlotStar_apply (ι : Fin 5 ↪ Fin (m+1))
(u : Fin (m+1) → CutRing × CutRing)
(F : OffsetFrame a r m hm (orderedTrackAlphabet ι) u) (V : polygonAlgebra a)
(s : UniversalExtension (alternatingGroup (Fin 5))) (j : Fin 5) (x : V.val) :
(coverMap M (alternatingGenerator a r m hm) (B.distinctSlotStar hlarge h hr ι u F V s)).val.val
(ι j,translate a (u (ι j)) x.val) =
(ι ((universalProjection _ s).val j),translate a (u (ι ((universalProjection _ s).val j))) x.val) := by
have he := B.frameStar_apply hlarge h hr (orderedTrackAlphabet ι) u F V
(universalMap (orderedTrackHom ι) s) ⟨ι j,orderedTrackAlphabet_mem ι j⟩ x
rw [show universalProjection _ (universalMap (orderedTrackHom ι) s) =
orderedTrackHom ι (universalProjection _ s) from
DFunLike.congr_fun (universalMap_spec (orderedTrackHom ι)) s,orderedTrackHom_apply] at he
exact he
theorem distinctSlotStar_projection (ι : Fin 5 ↪ Fin (m+1))
(u : Fin (m+1) → CutRing × CutRing)
(F : OffsetFrame a r m hm (orderedTrackAlphabet ι) u) (V : polygonAlgebra a) :
(coverMap M (alternatingGenerator a r m hm)).comp (B.distinctSlotStar hlarge h hr ι u F V) =
(slotAlternatingHom V (fun j => (ι j,u (ι j)))
(slots_injective_of_tracks V _ ι.injective)).comp (universalProjection (alternatingGroup (Fin 5))) := by
ext s : 1
apply Subtype.ext
apply Subtype.ext
apply Equiv.ext
intro p
by_cases hp : p ∈ Set.range (SlotMap a (m+1) V (fun j => (ι j,u (ι j))))
· obtain ⟨⟨j,x⟩,rfl⟩ := hp
change (coverMap M (alternatingGenerator a r m hm) (B.distinctSlotStar hlarge h hr ι u F V s)).val.val
(ι j,translate a (u (ι j)) x.val) =
slotPerm V (fun j => (ι j,u (ι j))) _ (universalProjection _ s).val
(SlotMap a (m+1) V (fun j => (ι j,u (ι j))) (j,x))
rw [slotPerm_apply]
exact B.distinctSlotStar_apply hlarge h hr ι u F V s j x
· have hp' : p ∉ frameDomain (orderedTrackAlphabet ι) u V := by
rintro ⟨i,hi,x,hx,rfl⟩
obtain ⟨j,hj,rfl⟩ := Finset.mem_map.mp hi
exact hp ⟨(j,⟨x,hx⟩),rfl⟩
have he := B.frameStar_supported hlarge h hr (orderedTrackAlphabet ι) u F V
(universalMap (orderedTrackHom ι) s) p hp'
change (coverMap M (alternatingGenerator a r m hm) (B.distinctSlotStar hlarge h hr ι u F V s)).val.val p =
slotPerm V (fun j => (ι j,u (ι j))) _ (universalProjection _ s).val p
rw [slotPerm_apply_of_notMem _ _ _ _ hp]
exact he
end InitialCoverSystem
end DistinctSlotStars
end SimpleAmenable
end
end
end OAI
|