openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /BurnsideDensity.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Function Module | |
| open scoped Classical | |
| namespace CriticalPercolation.LinearGrowth | |
| lemma toModuleEnd_surjective_of_simple {K A V : Type*} [Field K] [IsAlgClosed K] | |
| [Ring A] [Algebra K A] [AddCommGroup V] [Module K V] [Module A V] | |
| [IsScalarTower K A V] [IsSimpleModule A V] [FiniteDimensional K V] : | |
| Surjective (Module.toModuleEnd K (S:=A) V) := by | |
| let E := Module.End A V | |
| have he : Surjective (algebraMap K E) := | |
| (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed K).2 | |
| let : Module.Finite E V := Module.Finite.of_restrictScalars_finite K E V | |
| intro T | |
| let T' : Module.End E V := { | |
| toFun := T | |
| map_add' := T.map_add | |
| map_smul' := by | |
| intro e x | |
| obtain ⟨c,rfl⟩ := he e | |
| change T (c • x)=c • T x | |
| exact T.map_smul c x } | |
| obtain ⟨a,ha⟩ := Module.Finite.toModuleEnd_moduleEnd_surjective (R:=A) (M:=V) T' | |
| refine ⟨a,?_⟩ | |
| ext x | |
| exact congrArg (fun f : Module.End E V => f x) ha | |
| lemma subalgebra_eq_top_of_simple {K V : Type*} [Field K] [IsAlgClosed K] | |
| [AddCommGroup V] [Module K V] [FiniteDimensional K V] | |
| (A : Subalgebra K (Module.End K V)) | |
| [IsSimpleModule A V] : A=⊤ := by | |
| apply top_unique | |
| intro T _ | |
| obtain ⟨a,ha⟩ := toModuleEnd_surjective_of_simple (K:=K) (A:=A) (V:=V) T | |
| have he : (a : Module.End K V)=T := by | |
| ext v | |
| exact congrArg (fun f : Module.End K V => f v) ha | |
| exact he ▸ a.property | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |