openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /LocalPositiveSubspaceOrbit.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.LocalProjectiveAction | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Filter MeasureTheory NormedSpace Module | |
| open scoped Classical Topology ENNReal | |
| namespace CriticalPercolation.LocalProjective | |
| variable {K V : Type*} [NontriviallyNormedField K] [NormedAddCommGroup V] [NormedSpace K V] | |
| variable [CompleteSpace K] [FiniteDimensional K V] | |
| omit [CompleteSpace K] in | |
| lemma small_intersection_rank (U W : Submodule K V) | |
| (he : finrank K U=finrank K W) (hne : U≠W) : | |
| finrank K ↥(U⊓W)<finrank K U := by | |
| apply Submodule.finrank_lt_finrank_of_lt | |
| refine lt_of_le_of_ne inf_le_left ?_ | |
| intro hi | |
| have hle : U≤W := by rw [←hi]; exact inf_le_right | |
| exact hne (Submodule.eq_of_le_of_finrank_eq hle he) | |
| lemma finite_positive_subspaces (μ : Measure (Space K V)) [IsFiniteMeasure μ] | |
| (d : ℕ) (ε : ℝ≥0∞) (hε : ε≠0) | |
| (hnull : ∀ W : Submodule K V, finrank K W<d → μ (subspace W)=0) : | |
| {W : Submodule K V | finrank K W=d ∧ ε≤μ (subspace W)}.Finite := by | |
| let T := {W : Submodule K V | finrank K W=d ∧ ε≤μ (subspace W)} | |
| by_contra hfin | |
| have hinf : T.Infinite := hfin | |
| let : Infinite T := hinf.to_subtype | |
| have hdis : Pairwise (AEDisjoint μ on (fun W:T => subspace W.val)) := by | |
| intro U W hne | |
| change μ (subspace U.val∩subspace W.val)=0 | |
| rw [←subspace_inf] | |
| apply hnull | |
| rw [←U.property.1] | |
| apply small_intersection_rank _ _ (U.property.1.trans W.property.1.symm) | |
| exact fun he => hne (Subtype.ext he) | |
| have hs := tsum_measure_le_measure_univ | |
| (fun W:T => (measurable_subspace W.val).nullMeasurableSet) hdis | |
| have hc : (∑' _ : T, ε)≤∑' W:T, μ (subspace W.val) := | |
| ENNReal.tsum_le_tsum (fun W => W.property.2) | |
| rw [ENNReal.tsum_const_eq_top_of_ne_zero hε] at hc | |
| exact (measure_ne_top μ Set.univ) (top_unique (hc.trans hs)) | |
| omit [CompleteSpace K] in | |
| lemma exists_minimal_positive_subspace (μ : Measure (Space K V)) [IsProbabilityMeasure μ] | |
| (h : ∃ W : Submodule K V, W≠⊤ ∧ μ (subspace W)≠0) : | |
| ∃ W : Submodule K V, W≠⊥ ∧ W≠⊤ ∧ μ (subspace W)≠0 ∧ | |
| ∀ U : Submodule K V, finrank K U<finrank K W → μ (subspace U)=0 := by | |
| have hex : ∃ d:ℕ, ∃ W : Submodule K V, | |
| finrank K W=d ∧ W≠⊤ ∧ μ (subspace W)≠0 := by | |
| obtain ⟨W,hW,hμ⟩ := h | |
| exact ⟨finrank K W,W,rfl,hW,hμ⟩ | |
| obtain ⟨W,hWd,hWt,hWμ⟩ := Nat.find_spec hex | |
| refine ⟨W,?_,hWt,hWμ,?_⟩ | |
| · intro he | |
| exact hWμ (by simp [he]) | |
| · intro U hU | |
| by_contra hUμ | |
| have hut : U≠⊤ := by | |
| intro he | |
| have hh := Submodule.finrank_le W | |
| rw [he,finrank_top] at hU | |
| omega | |
| have hi := Nat.find_min' hex ⟨U,rfl,hut,hUμ⟩ | |
| omega | |
| lemma measure_subspace_map_eq (μ : Measure (Space K V)) | |
| (a : V ≃L[K] V) (hμ : Measure.map (action a) μ=μ) | |
| (W : Submodule K V) : | |
| μ (subspace (W.map a.toLinearEquiv.toLinearMap))=μ (subspace W) := by | |
| conv_lhs => rw [←hμ] | |
| rw [Measure.map_apply (continuous_action a).measurable | |
| (measurable_subspace _),preimage_action_subspace] | |
| lemma exists_finite_positive_subspace_orbit {G : Type*} [Group G] | |
| (ρ : G →* (V ≃L[K] V)) (μ : Measure (Space K V)) [IsProbabilityMeasure μ] | |
| (hμ : ∀ g, Measure.map (action (ρ g)) μ=μ) | |
| (h : ∃ W : Submodule K V, W≠⊤ ∧ μ (subspace W)≠0) : | |
| ∃ W : Submodule K V, W≠⊥ ∧ W≠⊤ ∧ | |
| (Set.range (fun g => W.map (ρ g).toLinearEquiv.toLinearMap)).Finite := by | |
| obtain ⟨W,hWb,hWt,hWμ,hnull⟩ := exists_minimal_positive_subspace μ h | |
| refine ⟨W,hWb,hWt,?_⟩ | |
| apply (finite_positive_subspaces μ (finrank K W) _ hWμ hnull).subset | |
| rintro U ⟨g,rfl⟩ | |
| refine ⟨(ρ g).toLinearEquiv.finrank_map_eq W,?_⟩ | |
| exact (measure_subspace_map_eq μ (ρ g) (hμ g) W).ge | |
| end CriticalPercolation.LocalProjective | |
| end | |
| end | |
| end OAI | |