openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /DistinctSlotStars.lean
| 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 | |