openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /CommonTranslatedCopies.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.ReserveAtom | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.InitialCoverSystem | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PrimitiveCopies | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section CommonTranslatedCopies | |
| structure CommonFrame (a : ℕ) (r : CutRing) (m : ℕ) (hm : 2 ≤ m) | |
| (I : Finset (Fin (m+1))) (u : CutRing × CutRing) where | |
| k : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) | |
| d : Fin (m+1) → CutRing × CutRing | |
| projection : sourceLatticeFullMap a r m hm k = trackTranslation d | |
| common : ∀ i ∈ I, d i = u | |
| noncomputable def commonFrame (a : ℕ) (r : CutRing) (m : ℕ) (hm : 2 ≤ m) | |
| (I : Finset (Fin (m+1))) (b : Fin (m+1)) (hb : b ∉ I) (u : CutRing × CutRing) : | |
| CommonFrame a r m hm I u := by | |
| let h := sourceLatticeFullMap_prescribed a r m hm I b hb (fun _ => u) | |
| exact { | |
| k := h.choose | |
| d := h.choose_spec.choose | |
| projection := h.choose_spec.choose_spec.1 | |
| common := h.choose_spec.choose_spec.2 } | |
| noncomputable def spatialTranslate {a : ℕ} (u : CutRing × CutRing) | |
| (U : polygonAlgebra a) : polygonAlgebra a := | |
| ⟨translate a (-u) ⁻¹' U.val,polygon_preimage_translate (-u) U.property⟩ | |
| namespace CommonFrame | |
| variable {a m : ℕ} {r : CutRing} {hm : 2 ≤ m} | |
| {I : Finset (Fin (m+1))} {u : CutRing × CutRing} | |
| (F : CommonFrame a r m hm I u) | |
| theorem conditional_projection (U : polygonAlgebra a) (s : alternatingGroup I) : | |
| sourceLatticeMap a r m hm F.k * conditionalAlternatingHom U (subtypeAlternatingHom I s) * | |
| (sourceLatticeMap a r m hm F.k)⁻¹ = | |
| conditionalAlternatingHom (spatialTranslate u U) (subtypeAlternatingHom I s) := by | |
| apply Subtype.ext | |
| change sourceLatticeFullMap a r m hm F.k * conditionalHom U (subtypeAlternatingHom I s).val * | |
| (sourceLatticeFullMap a r m hm F.k)⁻¹ = _ | |
| rw [F.projection] | |
| apply (conditional_translation_conjugate U (subtypeAlternatingHom I s).val F.d u ?_).symm | |
| intro i hi | |
| exact F.common i ((Equiv.Perm.mem_range_ofSubtype_iff.mp ⟨s.val,rfl⟩) | |
| (Equiv.Perm.mem_support.mpr hi)) | |
| end CommonFrame | |
| namespace InitialCoverSystem | |
| variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m} | |
| (B : InitialCoverSystem a r m hm M) | |
| theorem commonTranslatedAlphabet_projection {I : Finset (Fin (m+1))} | |
| {u : CutRing × CutRing} (F : CommonFrame a r m hm I u) (j : Fin 5) : | |
| (coverMap M (alternatingGenerator a r m hm)).comp (B.translatedAlphabet I j F.k) = | |
| (conditionalAlternatingHom (spatialTranslate u (initialTest a r j))).comp | |
| (subtypeAlternatingHom I) := by | |
| apply MonoidHom.ext | |
| intro s | |
| change coverMap M (alternatingGenerator a r m hm) | |
| (B.t F.k * B.initialConditional j (subtypeAlternatingHom I s) * (B.t F.k)⁻¹) = _ | |
| rw [map_mul,map_mul,map_inv] | |
| rw [show coverMap M (alternatingGenerator a r m hm) (B.t F.k) = | |
| sourceLatticeMap a r m hm F.k from DFunLike.congr_fun B.t_projection F.k] | |
| rw [show coverMap M (alternatingGenerator a r m hm) | |
| (B.initialConditional j (subtypeAlternatingHom I s)) = | |
| conditionalAlternatingHom (initialTest a r j) (subtypeAlternatingHom I s) from | |
| DFunLike.congr_fun (B.initialConditional_projection j) (subtypeAlternatingHom I s)] | |
| exact F.conditional_projection _ s | |
| theorem commonTranslatedAlphabet_independent {I : Finset (Fin (m+1))} | |
| {u v : CutRing × CutRing} (F : CommonFrame a r m hm I u) | |
| (G : CommonFrame a r m hm I v) (j : Fin 5) | |
| (huv : spatialTranslate u (initialTest a r j) = spatialTranslate v (initialTest a r j)) : | |
| B.translatedAlphabet I j F.k = B.translatedAlphabet I j G.k := by | |
| apply B.initialAlphabet_orbit_independent | |
| intro s | |
| rw [F.conditional_projection,G.conditional_projection,huv] | |
| noncomputable def primitiveCopy (I : Finset (Fin (m+1))) | |
| (b : Fin (m+1)) (hb : b ∉ I) (p : Fin 5 × (CutRing × CutRing)) : | |
| alternatingGroup I →* BoundedRelationCover M (alternatingGenerator a r m hm) := | |
| B.translatedAlphabet I p.1 (commonFrame a r m hm I b hb p.2).k | |
| theorem primitiveCopy_projection (I : Finset (Fin (m+1))) | |
| (b : Fin (m+1)) (hb : b ∉ I) (p : Fin 5 × (CutRing × CutRing)) : | |
| (coverMap M (alternatingGenerator a r m hm)).comp (B.primitiveCopy I b hb p) = | |
| (conditionalAlternatingHom (spatialTranslate p.2 (initialTest a r p.1))).comp | |
| (subtypeAlternatingHom I) := | |
| B.commonTranslatedAlphabet_projection (commonFrame a r m hm I b hb p.2) p.1 | |
| theorem primitiveCopy_aligned (I : Finset (Fin (m+1))) | |
| (b : Fin (m+1)) (hb : b ∉ I) (p : Fin 5 × (CutRing × CutRing)) (s : alternatingGroup I) : | |
| B.primitiveCopy I b hb p s ∈ sourceAlignedGroup a r m hm M B.t I := | |
| B.translatedAlphabet_aligned I p.1 _ s | |
| end InitialCoverSystem | |
| end CommonTranslatedCopies | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |