openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /AlignedFiniteTables.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.AlignedPolygonGeneration | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section AlignedFiniteTables | |
| variable {α G : Type*} [Group G] (f : α → G) | |
| theorem finite_table_lift_in_subgroup_eventually (L : ℕ) | |
| (K : Subgroup (BoundedRelationCover L f)) {T : Type*} [Group T] [Finite T] | |
| (φ : T →* G) (hφ : ∀ s, φ s ∈ K.map (coverMap L f)) : | |
| ∃ B : ℕ, ∃ hLB : L ≤ B, ∀ M : ℕ, ∀ hBM : B ≤ M, | |
| ∃ ρ : T →* BoundedRelationCover M f, | |
| (coverMap M f).comp ρ = φ ∧ | |
| ρ.range ≤ K.map (coverTransition f (le_trans hLB hBM)) := by | |
| choose y hy hyq using hφ | |
| choose w hw using fun s => PresentedGroup.mk_surjective (shortRelations L f) (y s) | |
| have hwp : ∀ s, FreeGroup.lift f (w s) = φ s := by | |
| intro s | |
| rw [← coverMap_mk L f,hw s,hyq s] | |
| obtain ⟨B,hB⟩ := finite_table_lift_eventually f φ w hwp | |
| refine ⟨max L B,le_max_left _ _,fun M hM => ?_⟩ | |
| obtain ⟨ρ,hρ,hρw⟩ := hB M (le_trans (le_max_right _ _) hM) | |
| refine ⟨ρ,hρ,?_⟩ | |
| rintro z ⟨s,rfl⟩ | |
| refine ⟨y s,hy s,?_⟩ | |
| rw [← hw s,coverTransition_mk,hρw s] | |
| end AlignedFiniteTables | |
| section SourceAlignedTableCompatibility | |
| variable (a : ℕ) (r : CutRing) (m : ℕ) (hm : 2 ≤ m) | |
| theorem sourceAlignedGroup_map_full (M : ℕ) | |
| (t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →* | |
| BoundedRelationCover M (alternatingGenerator a r m hm)) | |
| (ht : (coverMap M (alternatingGenerator a r m hm)).comp t = sourceLatticeMap a r m hm) | |
| (I : Finset (Fin (m+1))) : | |
| (sourceAlignedGroup a r m hm M t I).map | |
| ((polygonAlternatingGroup a (m+1)).subtype.comp | |
| (coverMap M (alternatingGenerator a r m hm))) = polygonAlignedGroup a r m hm I := by | |
| rw [sourceAlignedGroup,alignedGroup_map] | |
| have htrans : ((polygonAlternatingGroup a (m+1)).subtype.comp | |
| (coverMap M (alternatingGenerator a r m hm))).comp t = | |
| sourceLatticeFullMap a r m hm := by | |
| rw [MonoidHom.comp_assoc,ht] | |
| rfl | |
| rw [htrans] | |
| rfl | |
| theorem sourceAlignedGroup_transition {L M : ℕ} (hLM : L ≤ M) | |
| (t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →* | |
| BoundedRelationCover L (alternatingGenerator a r m hm)) (I : Finset (Fin (m+1))) : | |
| (sourceAlignedGroup a r m hm L t I).map | |
| (coverTransition (alternatingGenerator a r m hm) hLM) = | |
| sourceAlignedGroup a r m hm M | |
| ((coverTransition (alternatingGenerator a r m hm) hLM).comp t) I := by | |
| rw [sourceAlignedGroup,alignedGroup_map] | |
| rfl | |
| theorem fiveTrack_conditional_preimage_aligned | |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) (M : ℕ) | |
| (t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →* | |
| BoundedRelationCover M (alternatingGenerator a r m hm)) | |
| (ht : (coverMap M (alternatingGenerator a r m hm)).comp t = sourceLatticeMap a r m hm) | |
| (ι : Fin 5 ↪ Fin (m+1)) (b : Fin (m+1)) (hb : b ∉ Set.range ι) | |
| (U : polygonAlgebra a) (g : alternatingGroup (Fin 5)) : | |
| ∃ y ∈ sourceAlignedGroup a r m hm M t (Finset.univ.map ι), | |
| ((coverMap M (alternatingGenerator a r m hm) y).val : polygonFullGroup a (m+1)) = | |
| conditionalHom U (fiveTrackHom ι g) := by | |
| have hh := fiveTrack_polygon_aligned a r m hm hr ι b hb U g | |
| rw [← sourceAlignedGroup_map_full a r m hm M t ht] at hh | |
| exact hh | |
| end SourceAlignedTableCompatibility | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |