Statement.lean

-- Rellich-Kondrachov compactness (blocked: Sobolev spaces not in Mathlib)
-- Fallback: Arzelà-Ascoli for equicontinuous families
-- Status: ⚠️ partial (full theorem blocked by missing W^{1,p})

import Mathlib.Topology.UniformSpace.Ascoli
import Mathlib.Topology.ContinuousMap.Bounded.Basic

open BoundedContinuousFunction

/-- Arzelà-Ascoli style compactness: an equicontinuous, uniformly bounded
family of bounded continuous functions on a compact space is precompact.

This is a weaker substitute for the full Rellich-Kondrachov theorem,
which requires Sobolev spaces W^{1,p} not yet available in Mathlib. -/
theorem equicontinuous_precompact
    {X : Type*} [TopologicalSpace X] [CompactSpace X] [UniformSpace X]
    (F : Set (X →ᵇ ℝ))
    (hF_equicont : Equicontinuous (fun (f : ↥F) (x : X) => (f : X →ᵇ ℝ) x))
    (hF_bdd : ∃ M : ℝ, ∀ f ∈ F, ∀ x, |f x| ≤ M) :
    IsCompact (closure F) := by
  sorry