openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /FullSpanAlternative.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.FiniteSubspaceOrbit | |
| import OAI.Probability.CriticalPercolation.LinearGroups.LocalFurstenbergBound | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Module | |
| open scoped Classical | |
| namespace CriticalPercolation.LinearGrowth | |
| variable {K V G : Type*} [Field K] [AddCommGroup V] [Module K V] [Group G] | |
| def FullSpan (ρ : G →* (V ≃ₗ[K] V)) : Prop := | |
| Submodule.span K (Set.range (fun g => (ρ g).toLinearMap))=⊤ | |
| def StrongFullSpan (ρ : G →* (V ≃ₗ[K] V)) : Prop := | |
| ∀ H : Subgroup G, H.FiniteIndex → FullSpan (ρ.comp H.subtype) | |
| lemma submodule_eq_bot_or_top_of_fullSpan (ρ : G →* (V ≃ₗ[K] V)) (hfull : FullSpan ρ) | |
| (W : Submodule K V) (hinv : ∀ g, W.map (ρ g).toLinearMap=W) : W=⊥ ∨ W=⊤ := by | |
| by_cases hW : W=⊥ | |
| · exact Or.inl hW | |
| right | |
| obtain ⟨w,hw,hw0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hW | |
| obtain ⟨φ,hφ⟩ := Projective.exists_dual_eq_one K hw0 | |
| apply top_unique | |
| intro y _ | |
| let T := φ.smulRight y | |
| have hT : T∈Submodule.span K (Set.range (fun g => (ρ g).toLinearMap)) := hfull ▸ Submodule.mem_top | |
| have hm : ∀ A∈Submodule.span K (Set.range (fun g => (ρ g).toLinearMap)), A w∈W := by | |
| intro A hA | |
| induction hA using Submodule.span_induction with | |
| | mem A hA => | |
| obtain ⟨g,rfl⟩ := hA | |
| rw [←hinv g] | |
| exact Submodule.mem_map.mpr ⟨w,hw,rfl⟩ | |
| | zero => exact W.zero_mem | |
| | add A B hA hB ihA ihB => exact W.add_mem ihA ihB | |
| | smul c A hA ihA => exact W.smul_mem c ihA | |
| simpa only [T,LinearMap.smulRight_apply,hφ,one_smul] using hm T hT | |
| lemma no_finite_subspace_orbit_of_strongFullSpan (ρ : G →* (V ≃ₗ[K] V)) | |
| (hρ : StrongFullSpan ρ) (W : Submodule K V) (hW0 : W≠⊥) (hW1 : W≠⊤) : | |
| ¬(Set.range (fun g => W.map (ρ g).toLinearMap)).Finite := by | |
| intro hf | |
| obtain ⟨H,hHi,hH⟩ := finiteIndex_subspace_stabilizer ρ W hf | |
| exact (submodule_eq_bot_or_top_of_fullSpan (ρ.comp H.subtype) (hρ H hHi) W hH).elim hW0 hW1 | |
| end CriticalPercolation.LinearGrowth | |
| namespace CriticalPercolation.LinearGrowth | |
| variable {K V G : Type*} [NontriviallyNormedField K] [NormedAddCommGroup V] | |
| variable [NormedSpace K V] [ProperSpace K] [FiniteDimensional K V] [Group G] | |
| def continuousRepresentation (ρ : G →* (V ≃ₗ[K] V)) : G →* (V ≃L[K] V) where | |
| toFun g := (ρ g).toContinuousLinearEquiv | |
| map_one' := by ext x; change ρ 1 x=x; rw [map_one]; rfl | |
| map_mul' g h := by ext x; change ρ (g*h) x=ρ g (ρ h x); rw [map_mul]; rfl | |
| lemma projective_bound_of_strongFullSpan [Nontrivial V] [DecidableEq G] | |
| (ρ : G →* (V ≃ₗ[K] V)) (hρ : StrongFullSpan ρ) | |
| (S : Finset G) (h1 : (1:G)∈S) (hgen : Subgroup.closure (S:Set G)=⊤) | |
| (d : ℕ) (hg : SparseGrowth.HasDegree S d) : | |
| ∃ δ : ℝ, 0<δ ∧ ∀ g : G, ∀ x : V, | |
| δ*‖(continuousRepresentation ρ g).toContinuousLinearMap‖*‖x‖≤‖ρ g x‖ := by | |
| rcases LocalProjective.sparse_local_furstenberg_alternative (continuousRepresentation ρ) | |
| S h1 hgen d hg with horb | hb | |
| · obtain ⟨W,hW0,hW1,hW⟩ := horb | |
| exact False.elim (no_finite_subspace_orbit_of_strongFullSpan ρ hρ W hW0 hW1 hW) | |
| · exact hb | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |