Prove2Me

`PhysFunctionalAnalysis.countable_of_separated` {X : Type*} [PseudoMetricSpace X] [TopologicalSpace.SeparableSpace X] {ι : Type*} (u : ι → X) {r : ℝ} (hr : 0 < r) (hsep : ∀ i j, i

Proved
PhysFunctionalAnalysis.countable_of_separated

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from PhysFunctionalAnalysis.

PhysFunctionalAnalysis.countable_of_separated {X : Type*} [PseudoMetricSpace X] [TopologicalSpace.SeparableSpace X] {ι : Type*} (u : ι → X) {r : ℝ} (hr : 0 < r) (hsep : ∀ i j, i ≠ j → r ≤ dist (u i) (u j)) : Countable ι

Formalization note: Lean 4 identifier PhysFunctionalAnalysis.countable_of_separated.

Preamble
-- Generated from PhysFunctionalAnalysis.lean — theorem PhysFunctionalAnalysis.countable_of_separated
import Mathlib
import Definitions.Def_PhysFunctionalAnalysis
import Definitions.Def_ChapterA4
open PhysFunctionalAnalysis


open MeasureTheory Set
open scoped ENNReal lp

noncomputable section


open PhysMeasureBasis
Formal statement
theorem PhysFunctionalAnalysis.countable_of_separated {X : Type*} [PseudoMetricSpace X]
    [TopologicalSpace.SeparableSpace X] {ι : Type*} (u : ι → X) {r : ℝ} (hr : 0 < r)
    (hsep : ∀ i j, i ≠ j → r ≤ dist (u i) (u j)) : Countable ι := by sorry

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