openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /CompactSmallObservations.lean
| import OAI.NumberTheory.DukePrimeDegree.Geometry.QuotientNeighborhoods | |
| import OAI.NumberTheory.DukePrimeDegree.Geometry.QuotientSeparatingCuts | |
| namespace OAI | |
| /-! # Finite observations with uniformly small lifts on a compact set -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| theorem exists_finite_observation_small_lifts {n : ℕ} | |
| (K : Set (X n)) (hK : IsCompact K) (W : Set (G n)) (hW : W ∈ 𝓝 1) : | |
| ∃ (N : ℕ) (Q : X n → Fin N), Measurable Q ∧ | |
| ∀ x ∈ K, ∀ y, Q y=Q x → ∃ g ∈ W, rightAction x g=y := by | |
| classical | |
| choose U hU hxU hsmall using fun x : X n => exists_open_small_relative_lifts x W hW | |
| obtain ⟨s,hs⟩ := hK.elim_finite_subcover U hU (fun x _ => mem_iUnion.mpr ⟨x,hxU x⟩) | |
| let bits : X n → (s → Bool) := fun x i => decide (x ∈ U i.val) | |
| have hbits : Measurable bits := by | |
| apply measurable_pi_iff.mpr | |
| intro i | |
| apply measurable_to_bool | |
| have he : (fun x => bits x i) ⁻¹' {true} = U i.val := by | |
| ext x | |
| simp [bits] | |
| rw [he] | |
| exact (hU i.val).measurableSet | |
| let Q : X n → Fin (Fintype.card (s → Bool)) := fun x => Fintype.equivFin _ (bits x) | |
| refine ⟨_,Q,(measurable_of_finite (Fintype.equivFin _)).comp hbits,?_⟩ | |
| intro x hx y heq | |
| obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (hs hx) | |
| have hxy : bits y=bits x := (Fintype.equivFin _).injective heq | |
| have hbit := congrFun hxy ⟨i,hi⟩ | |
| have hyi : y ∈ U i := by | |
| simpa only [bits,hxi,decide_true,decide_eq_true_eq] using hbit | |
| exact hsmall i x hxi y hyi | |
| end DukePrimeDegree | |
| end | |
| end OAI | |