openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /PolygonAlternatingConfigurations.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.TransitiveInduction | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.PolygonBankEmbedding | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonAlternatingConfigurations | |
| open Classical CategoryTheory Set | |
| namespace PolygonPlacement | |
| variable {a m n : ℕ} | |
| noncomputable def bankEmbedding (f : Fin n → PolygonPlacement a m) | |
| (hf : Pairwise (fun i j => Apart (f i) (f j))) : | |
| PolygonObject.BankEmbedding (PolygonObject.whole a n) m where | |
| emb := ⟨fun x => bank f hf x.val,fun x y h => Subtype.ext ((bank f hf).injective h)⟩ | |
| table := by | |
| choose S hS hcov using fun i => (f i).charts | |
| let c (i : Fin n) (k : Fin m × (CutRing×CutRing) × polygonAlgebra a) : | |
| PolygonObject.Chart (a:=a) n m := ⟨i,k.1,k.2.1,k.2.2⟩ | |
| refine ⟨Finset.univ.biUnion (fun i => (S i).image (c i)),?_,?_⟩ | |
| · intro k hk | |
| obtain ⟨i,_,hk⟩ := Finset.mem_biUnion.mp hk | |
| obtain ⟨d,hd,rfl⟩ := Finset.mem_image.mp hk | |
| intro z hz | |
| exact ⟨by trivial,hS i d hd z hz⟩ | |
| · intro x | |
| obtain ⟨d,hd,hz⟩ := hcov x.val.1 x.val.2 | |
| exact ⟨c x.val.1 d,Finset.mem_biUnion.mpr ⟨x.val.1,Finset.mem_univ _, | |
| Finset.mem_image.mpr ⟨d,hd,rfl⟩⟩,rfl,hz⟩ | |
| @[simp] theorem bankEmbedding_apply (f : Fin n → PolygonPlacement a m) | |
| (hf : Pairwise (fun i j => Apart (f i) (f j))) | |
| (x : (PolygonObject.whole a n).Point) : bankEmbedding f hf x=f x.val.1 x.val.2 := rfl | |
| theorem alternating_disjoint_configurations (f g : Fin n → PolygonPlacement a m) | |
| (hf : Pairwise (fun i j => Apart (f i) (f j))) | |
| (hg : Pairwise (fun i j => Apart (g i) (g j))) | |
| (hfg : ∀i j,Apart (f i) (g j)) (hm : 3*n+1 < m) : | |
| ∃c : polygonAlternatingGroup a m,∀i x,c.val.val (f i x)=g i x := by | |
| let S := Finset.univ.image f ∪ Finset.univ.image g | |
| have hS : S.card≤2*n := by | |
| calc | |
| _≤(Finset.univ.image f).card+(Finset.univ.image g).card := Finset.card_union_le _ _ | |
| _≤(Finset.univ : Finset (Fin n)).card+(Finset.univ : Finset (Fin n)).card := | |
| Nat.add_le_add Finset.card_image_le Finset.card_image_le | |
| _=2*n := by simp; omega | |
| obtain ⟨w,hw,hws⟩ := exists_configuration_avoiding S n (by omega) | |
| have hwf (i j : Fin n) : Apart (w i) (f j) := | |
| hws i (f j) (Finset.mem_union_left _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩)) | |
| have hwg (i j : Fin n) : Apart (w i) (g j) := | |
| hws i (g j) (Finset.mem_union_right _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩)) | |
| obtain ⟨c,hc,_⟩ := PolygonObject.BankEmbedding.cycle_three | |
| (bankEmbedding f hf) (bankEmbedding g hg) (bankEmbedding w hw) | |
| (fun p q => (apart_iff _ _).mp (hfg p.val.1 q.val.1) p.val.2 q.val.2) | |
| (fun p q => (apart_iff _ _).mp (hwf q.val.1 p.val.1).symm p.val.2 q.val.2) | |
| (fun p q => (apart_iff _ _).mp (hwg q.val.1 p.val.1).symm p.val.2 q.val.2) | |
| exact ⟨c,fun i x => hc ⟨(i,x),by trivial⟩⟩ | |
| theorem alternating_configuration_transitive_sharp (f g : Fin n → PolygonPlacement a m) | |
| (hf : Pairwise (fun i j => Apart (f i) (f j))) | |
| (hg : Pairwise (fun i j => Apart (g i) (g j))) (hm : 3*n+1 < m) : | |
| ∃c : polygonAlternatingGroup a m,∀i,c.val • f i=g i := by | |
| let S := Finset.univ.image f ∪ Finset.univ.image g | |
| have hS : S.card≤2*n := by | |
| calc | |
| _≤(Finset.univ.image f).card+(Finset.univ.image g).card := Finset.card_union_le _ _ | |
| _≤(Finset.univ : Finset (Fin n)).card+(Finset.univ : Finset (Fin n)).card := | |
| Nat.add_le_add Finset.card_image_le Finset.card_image_le | |
| _=2*n := by simp; omega | |
| obtain ⟨w,hw,hws⟩ := exists_configuration_avoiding S n (by omega) | |
| have hwf (i j : Fin n) : Apart (w i) (f j) := | |
| hws i (f j) (Finset.mem_union_left _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩)) | |
| have hwg (i j : Fin n) : Apart (w i) (g j) := | |
| hws i (g j) (Finset.mem_union_right _ (Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩)) | |
| obtain ⟨u,hu⟩ := alternating_disjoint_configurations f w hf hw (fun i j => (hwf j i).symm) hm | |
| obtain ⟨v,hv⟩ := alternating_disjoint_configurations w g hw hg hwg hm | |
| refine ⟨v*u,fun i => ?_⟩ | |
| apply PolygonPlacement.ext | |
| intro x | |
| change v.val.val (u.val.val (f i x))=g i x | |
| rw [hu,hv] | |
| namespace Configuration | |
| theorem alternating_transitive {p : ℕ} (f g : Configuration a m p) (hm : 3*p+1 < m) : | |
| ∃c : polygonAlternatingGroup a m,c.val • f=g := by | |
| obtain ⟨c,hc⟩ := alternating_configuration_transitive_sharp f.val g.val f.property g.property hm | |
| exact ⟨c,ext hc⟩ | |
| theorem alternating_stabilizer_standard (a p n : ℕ) : | |
| MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)= | |
| (PolygonTracks.bankFixer a p n).comap (polygonAlternatingGroup a (p+n)).subtype := by | |
| ext c | |
| change c.val∈MulAction.stabilizer (polygonFullGroup a (p+n)) (standard a p n) ↔ | |
| c.val∈PolygonTracks.bankFixer a p n | |
| rw [stabilizer_standard] | |
| noncomputable def alternatingOrbitHomologyIso (a p n : ℕ) (hm : 3*p+1 < p+n) (q : ℕ) : | |
| groupHomology (Rep.of (TransitiveInduction.permutationRep | |
| (G:=polygonAlternatingGroup a (p+n)) (X:=Configuration a (p+n) p))) q ≅ | |
| groupHomology (Rep.trivial ℤ (MulAction.stabilizer (polygonAlternatingGroup a (p+n)) | |
| (standard a p n)) ℤ) q := | |
| TransitiveInduction.homologyIso (standard a p n) (fun f => alternating_transitive _ f hm) q | |
| noncomputable def alternatingRepresentationIso (a m p : ℕ) : | |
| Rep.of (TransitiveInduction.permutationRep | |
| (G:=polygonAlternatingGroup a m) (X:=Configuration a m p)) ≅ | |
| Rep.of ((configurationRepresentation a m p).comp (polygonAlternatingGroup a m).subtype) := | |
| Rep.mkIso | |
| { toLinearEquiv := chainEquiv a m p | |
| isIntertwining' := fun g => chainEquiv_intertwines g.val } | |
| noncomputable def alternatingColumnHomologyIso (a p n : ℕ) (hm : 3*p+1 < p+n) (q : ℕ) : | |
| groupHomology (Rep.of ((configurationRepresentation a (p+n) p).comp | |
| (polygonAlternatingGroup a (p+n)).subtype)) q ≅ | |
| groupHomology (Rep.trivial ℤ (MulAction.stabilizer (polygonAlternatingGroup a (p+n)) | |
| (standard a p n)) ℤ) q := | |
| (groupHomology.functor ℤ _ q).mapIso (alternatingRepresentationIso a (p+n) p).symm ≪≫ | |
| alternatingOrbitHomologyIso a p n hm q | |
| end Configuration | |
| end PolygonPlacement | |
| end PolygonAlternatingConfigurations | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |