openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Homology /ConfigurationResolution.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonFixedTrackProgress | |
| import OAI.GroupTheory.SimpleAmenable.Homology.ConfigurationEquivariance | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section ConfigurationResolution | |
| open Classical CategoryTheory Finsupp | |
| namespace ConfigurationChains | |
| variable {V : Type*} {R : V → V → Prop} | |
| noncomputable def complex (R : V → V → Prop) : ChainComplex (ModuleCat ℤ) ℕ := | |
| ChainComplex.of (fun n => ModuleCat.of ℤ (degree R n)) | |
| (fun n => ModuleCat.ofHom (degreeBoundary R n)) (fun n => by | |
| apply ModuleCat.hom_ext | |
| apply LinearMap.ext | |
| intro c | |
| exact degreeBoundary_square R n c) | |
| noncomputable def representation {G : Type*} [Monoid G] [MulAction G V] | |
| (hR : ∀g : G,∀v w,R v w → R (g • v) (g • w)) (n : ℕ) : | |
| Representation ℤ G (degree R n) where | |
| toFun g := degreePush (fun v => g • v) (hR g) n | |
| map_one' := by | |
| apply LinearMap.ext | |
| intro c | |
| apply Subtype.ext | |
| change push (fun v : V => (1:G) • v) c.val=c.val | |
| have he : (fun v : V => (1:G) • v)=id := funext (fun v => one_smul G v) | |
| rw [he] | |
| exact push_id c.val | |
| map_mul' g h := by | |
| apply LinearMap.ext | |
| intro c | |
| apply Subtype.ext | |
| change push (fun v : V => (g*h) • v) c.val=push (fun v => g • v) (push (fun v => h • v) c.val) | |
| rw [push_comp] | |
| simp only [Function.comp_def, mul_smul] | |
| @[simp] theorem representation_boundary {G : Type*} [Monoid G] [MulAction G V] | |
| (hR : ∀g : G,∀v w,R v w → R (g • v) (g • w)) (g : G) (n : ℕ) | |
| (c : degree R (n+1)) : | |
| representation hR n g (degreeBoundary R n c)= | |
| degreeBoundary R n (representation hR (n+1) g c) := | |
| degreePush_boundary _ (hR g) n c | |
| noncomputable def simplexModuleEquiv (R : V → V → Prop) (n : ℕ) : | |
| degree R n ≃ₗ[ℤ] ({l : List V // l.Pairwise R ∧ l.length=n} →₀ ℤ) := by | |
| have he : degree R n=Finsupp.supported ℤ ℤ {l : List V | l.Pairwise R ∧ l.length=n} := by | |
| ext c | |
| rw [Finsupp.mem_supported] | |
| rfl | |
| exact (LinearEquiv.ofEq _ _ he).trans (Finsupp.supportedEquivFinsupp _) | |
| theorem lowDegree_exact | |
| (hc : ∀S : Finset V,S.card≤31 → ∃v,∀w∈S,R v w) | |
| {n : ℕ} (hn : n≤2) : | |
| Function.Exact (degreeBoundary R (n+1)) (degreeBoundary R n) := by | |
| intro c | |
| constructor | |
| · intro hz | |
| have hz' : boundary c.val=0 := congrArg Subtype.val hz | |
| obtain ⟨d,hd,he⟩ := low_degree_filling hc (by omega : n+1≤3) c.property hz' | |
| exact ⟨⟨d,hd⟩,Subtype.ext he⟩ | |
| · rintro ⟨d,rfl⟩ | |
| exact degreeBoundary_square R n d | |
| end ConfigurationChains | |
| namespace PolygonPlacement | |
| noncomputable def configurationRepresentation (a m n : ℕ) : | |
| Representation ℤ (polygonFullGroup a m) (ConfigurationChains.degree (@Apart a m) n) := | |
| ConfigurationChains.representation (fun g v w h => (apart_smul_iff g v w).mpr h) n | |
| end PolygonPlacement | |
| end ConfigurationResolution | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |