openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /RealVirtualSolvability.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.AlgebraicVirtualSolvability | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Module | |
| open scoped Classical Cardinal | |
| namespace CriticalPercolation.LinearGrowth | |
| lemma countable_finiteType_int (D : Type) [CommRing D] [Algebra ℤ D] | |
| [Algebra.FiniteType ℤ D] : Countable D := by | |
| obtain ⟨n,f,hf⟩ := (Algebra.FiniteType.iff_quotient_mvPolynomial).mp | |
| (inferInstance : Algebra.FiniteType ℤ D) | |
| let : Countable (MvPolynomial n ℤ) := by | |
| apply Function.Injective.countable (f := AddMonoidAlgebra.coeff) | |
| intro x y h | |
| cases x; cases y; cases h; rfl | |
| exact Function.Surjective.countable hf | |
| lemma countable_fraction (D : Type) [CommRing D] [Countable D] : Countable (FractionRing D) := by | |
| apply Cardinal.mk_le_aleph0_iff.mp | |
| rw [Cardinal.mk_fractionRing] | |
| exact Cardinal.mk_le_aleph0 | |
| lemma countable_algebraicClosure (F : Type) [Field F] [Countable F] : | |
| Countable (AlgebraicClosure F) := by | |
| have h := Algebraic.countable F (AlgebraicClosure F) | |
| have he : {x : AlgebraicClosure F | IsAlgebraic F x}=Set.univ := | |
| Set.eq_univ_of_forall (fun _ => Algebra.IsAlgebraic.isAlgebraic _) | |
| rw [he] at h | |
| exact Set.countable_univ_iff.mp h | |
| lemma matrixRepresentation_injective {K G n : Type} [Field K] [Group G] | |
| [Fintype n] [DecidableEq n] (ρ : G →* GL n K) (hρ : Injective ρ) : | |
| Injective (matrixRepresentation ρ) := | |
| (LinearMap.GeneralLinearGroup.generalLinearEquiv K (n → K)).injective.comp | |
| (Matrix.GeneralLinearGroup.toLin.injective.comp hρ) | |
| lemma virtually_solvable_real_matrix {G n : Type} [Group G] [DecidableEq G] | |
| [Fintype n] [DecidableEq n] (ρ : G →* GL n ℝ) (hρ : Injective ρ) | |
| {degree : ℕ} (hg : SparseGrowth.HasPolynomialDegree G degree) : | |
| ∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H := by | |
| let := hg.fg | |
| obtain ⟨D,hD,hm⟩ := exists_finiteType_matrix_ring ρ | |
| let := hD | |
| let : Countable D := countable_finiteType_int D | |
| let F := FractionRing D | |
| let K := AlgebraicClosure F | |
| let : Countable F := countable_fraction D | |
| let : Countable K := countable_algebraicClosure F | |
| let f : D →+* K := (algebraMap F K).comp (algebraMap D F) | |
| have hf : Injective f := (algebraMap F K).injective.comp (IsFractionRing.injective D F) | |
| let σ := liftMatrixGroup ρ D hm | |
| let τ := (Matrix.GeneralLinearGroup.map f).comp σ | |
| have hσ : Injective σ := liftMatrixGroup_injective ρ D hm hρ | |
| have hτ : Injective τ := by | |
| intro x y h | |
| apply hσ | |
| apply Units.ext | |
| apply Matrix.ext | |
| intro i j | |
| apply hf | |
| exact congrArg (fun M : GL n K => M i j) h | |
| exact virtually_solvable_linear K (n → K) G (matrixRepresentation τ) | |
| (matrixRepresentation_injective τ hτ) hg | |
| lemma virtually_solvable_real_linear {G V : Type} [Group G] [DecidableEq G] | |
| [AddCommGroup V] [Module ℝ V] [FiniteDimensional ℝ V] | |
| (ρ : G →* (V ≃ₗ[ℝ] V)) (hρ : Injective ρ) | |
| {degree : ℕ} (hg : SparseGrowth.HasPolynomialDegree G degree) : | |
| ∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H := by | |
| let τ := (linearMatrixEquiv (K:=ℝ) (V:=V)).toMonoidHom.comp ρ | |
| exact virtually_solvable_real_matrix τ (linearMatrixEquiv.injective.comp hρ) hg | |
| end CriticalPercolation.LinearGrowth | |
| namespace CriticalPercolation.LinearGrowth | |
| lemma virtually_solvable_real_linear_general {G V : Type*} [Group G] [DecidableEq G] | |
| [AddCommGroup V] [Module ℝ V] [FiniteDimensional ℝ V] | |
| (ρ : G →* (V ≃ₗ[ℝ] V)) (hρ : Injective ρ) | |
| {degree : ℕ} (hg : SparseGrowth.HasPolynomialDegree G degree) : | |
| ∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H := by | |
| let τ := (linearMatrixEquiv (K:=ℝ) (V:=V)).toMonoidHom.comp ρ | |
| have hτ : Injective τ := linearMatrixEquiv.injective.comp hρ | |
| have hgτ := hg.of_surjective τ.rangeRestrict τ.rangeRestrict_surjective | |
| obtain ⟨P,hPi,hPs⟩ := virtually_solvable_real_matrix τ.range.subtype Subtype.val_injective hgτ | |
| let := hPi | |
| let := hPs | |
| let H := P.comap τ.rangeRestrict | |
| have hHi : H.FiniteIndex := finiteIndex_comap_surjective τ.rangeRestrict τ.rangeRestrict_surjective P | |
| let ψ : H →* P := (τ.rangeRestrict.comp H.subtype).codRestrict P (fun h => h.property) | |
| have hψ : Injective ψ := by | |
| intro x y he | |
| apply Subtype.ext | |
| apply hτ | |
| exact congrArg (fun z : P => (z.val : GL (Fin (finrank ℝ V)) ℝ)) he | |
| exact ⟨H,hHi,Group.isSolvable_of_isSolvable_injective hψ⟩ | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |