openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Homology /FinitePrimitiveResolution.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ResolvedBoolean | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section FinitePrimitiveResolution | |
| variable {X ι : Type*} | |
| theorem finite_resolving_subfamily (U : ι → Set X) {V : Set X} | |
| (hV : V ∈ BooleanSubalgebra.closure (Set.range U)) : | |
| ∃ S : Finset ι, ResolvedBy (fun i : S => U i.val) V := by | |
| classical | |
| induction hV using BooleanSubalgebra.closure_bot_sup_induction with | |
| | mem V h => | |
| obtain ⟨i,rfl⟩ := h | |
| exact ⟨{i},fun x y he => he ⟨i,Finset.mem_singleton_self i⟩⟩ | |
| | bot => exact ⟨∅,fun _ _ _ => Iff.rfl⟩ | |
| | sup V W _ _ ihV ihW => | |
| obtain ⟨S,hS⟩ := ihV | |
| obtain ⟨T,hT⟩ := ihW | |
| refine ⟨S ∪ T,fun x y he => or_congr (hS x y ?_) (hT x y ?_)⟩ | |
| · intro i | |
| exact he ⟨i.val,Finset.mem_union_left _ i.property⟩ | |
| · intro i | |
| exact he ⟨i.val,Finset.mem_union_right _ i.property⟩ | |
| | compl V _ ih => | |
| obtain ⟨S,hS⟩ := ih | |
| exact ⟨S,fun x y he => not_congr (hS x y he)⟩ | |
| noncomputable def translatedPrimitiveAlgebra (a : ℕ) (r : CutRing) : | |
| BooleanSubalgebra (Set (GenericSquare a)) := | |
| BooleanSubalgebra.closure (Set.range fun p : Fin 5 × (CutRing × CutRing) => | |
| (spatialTranslate p.2 (initialTest a r p.1)).val) | |
| theorem translatedPrimitiveAlgebra_test (a : ℕ) (r : CutRing) | |
| (p : Fin 5 × (CutRing × CutRing)) : | |
| (spatialTranslate p.2 (initialTest a r p.1)).val ∈ translatedPrimitiveAlgebra a r := | |
| BooleanSubalgebra.subset_closure ⟨p,rfl⟩ | |
| theorem translatedPrimitiveAlgebra_translate (a : ℕ) (r : CutRing) | |
| (u : CutRing × CutRing) {V : Set (GenericSquare a)} | |
| (hV : V ∈ translatedPrimitiveAlgebra a r) : | |
| translate a u ⁻¹' V ∈ translatedPrimitiveAlgebra a r := by | |
| induction hV using BooleanSubalgebra.closure_bot_sup_induction with | |
| | mem V h => | |
| obtain ⟨p,rfl⟩ := h | |
| have he : translate a u ⁻¹' (spatialTranslate p.2 (initialTest a r p.1)).val = | |
| (spatialTranslate (p.2-u) (initialTest a r p.1)).val := by | |
| ext x | |
| simp only [spatialTranslate,Set.mem_preimage,← translate_add] | |
| abel_nf | |
| rw [he] | |
| exact translatedPrimitiveAlgebra_test a r (p.1,p.2-u) | |
| | bot => exact BooleanSubalgebra.bot_mem | |
| | sup V W _ _ ihV ihW => exact BooleanSubalgebra.sup_mem ihV ihW | |
| | compl V _ ih => exact BooleanSubalgebra.compl_mem ih | |
| theorem translatedPrimitiveAlgebra_initial (a : ℕ) (r : CutRing) (j : Fin 5) : | |
| (initialTest a r j).val ∈ translatedPrimitiveAlgebra a r := by | |
| have h := translatedPrimitiveAlgebra_test a r (j,0) | |
| have he : (spatialTranslate (0:CutRing × CutRing) (initialTest a r j)).val = | |
| (initialTest a r j).val := by | |
| ext p | |
| simp [spatialTranslate,translate_zero] | |
| rwa [he] at h | |
| theorem polygon_finite_primitive_resolution {a : ℕ} (r : CutRing) | |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) (V : polygonAlgebra a) : | |
| ∃ S : Finset (Fin 5 × (CutRing × CutRing)), | |
| ResolvedBy (fun p : S => (spatialTranslate p.val.2 (initialTest a r p.val.1)).val) V.val := by | |
| apply finite_resolving_subfamily (fun p : Fin 5 × (CutRing × CutRing) => | |
| (spatialTranslate p.2 (initialTest a r p.1)).val) | |
| let B := translatedPrimitiveAlgebra a r | |
| have ht := fun u U hU => translatedPrimitiveAlgebra_translate a r u (V := U) hU | |
| have hcoord (j : Fin 2) (z : CutRing) : | |
| halfPlane a (Fin.castLE (by omega) j) z ∈ B := by | |
| apply coordinate_cuts_generated B j ht | |
| fin_cases j | |
| · exact translatedPrimitiveAlgebra_initial a r 1 | |
| · exact translatedPrimitiveAlgebra_initial a r 2 | |
| apply polygon_induction (P := fun U => U ∈ B) ?_ BooleanSubalgebra.bot_mem | |
| (fun _ _ hU hV => BooleanSubalgebra.sup_mem hU hV) | |
| (fun _ hU => BooleanSubalgebra.compl_mem hU) V.property | |
| intro j z | |
| fin_cases j | |
| · exact hcoord 0 z | |
| · exact hcoord 1 z | |
| · exact slope_cuts_generated r hr B ht hcoord 2 (Or.inl rfl) | |
| (translatedPrimitiveAlgebra_initial a r 3) z | |
| · exact slope_cuts_generated r hr B ht hcoord 3 (Or.inr rfl) | |
| (translatedPrimitiveAlgebra_initial a r 4) z | |
| end FinitePrimitiveResolution | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |