| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Homology.FinitePrimitiveResolution |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section FiniteCutResolution |
|
|
| theorem finite_cut_resolution {a : ℕ} {κ : Type*} [Finite κ] (R : κ → polygonAlgebra a) : |
| ∃ S : Finset (Fin 4 × CutRing), ∀ i, |
| ResolvedBy (fun p : S => halfPlane a p.val.1 p.val.2) (R i).val := by |
| classical |
| let _ := Fintype.ofFinite κ |
| have hgen : {U : Set (GenericSquare a) | ∃ j z, U=halfPlane a j z}= |
| Set.range (fun p : Fin 4 × CutRing => halfPlane a p.1 p.2) := by |
| ext U |
| constructor |
| · rintro ⟨j,z,rfl⟩; exact ⟨(j,z),rfl⟩ |
| · rintro ⟨⟨j,z⟩,rfl⟩; exact ⟨j,z,rfl⟩ |
| have h (i : κ) : ∃ S : Finset (Fin 4 × CutRing), |
| ResolvedBy (fun p : S => halfPlane a p.val.1 p.val.2) (R i).val := by |
| apply finite_resolving_subfamily (fun p : Fin 4 × CutRing => halfPlane a p.1 p.2) |
| have hp := (R i).property |
| change (R i).val ∈ BooleanSubalgebra.closure _ at hp |
| rwa [hgen] at hp |
| choose S hS using h |
| refine ⟨Finset.univ.biUnion S,fun i x y hh => ?_⟩ |
| apply hS i x y |
| intro p |
| exact hh ⟨p.val,Finset.mem_biUnion.mpr ⟨i,Finset.mem_univ i,p.property⟩⟩ |
|
|
| theorem finite_cut_resolution_extend {a : ℕ} {κ η : Type*} [Finite κ] [Finite η] |
| (R : κ → polygonAlgebra a) (j : η → Fin 4) (c : η → CutRing) : |
| ∃ S : Finset (Fin 4 × CutRing), |
| (∀ i, ResolvedBy (fun p : S => halfPlane a p.val.1 p.val.2) (R i).val) ∧ |
| ∀ e, (j e,c e) ∈ S := by |
| classical |
| let _ := Fintype.ofFinite η |
| obtain ⟨S,hS⟩ := finite_cut_resolution R |
| refine ⟨S ∪ Finset.univ.image (fun e => (j e,c e)),?_,?_⟩ |
| · intro i x y hh |
| exact hS i x y (fun p => hh ⟨p.val,Finset.mem_union_left _ p.property⟩) |
| · intro e |
| exact Finset.mem_union_right _ (Finset.mem_image.mpr ⟨e,Finset.mem_univ e,rfl⟩) |
|
|
| end FiniteCutResolution |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
|
|
