/-- Let `X` be a locally compact sigma compact Hausdorff topological space, let `s` be a closed set
in `X`. Suppose that for each `x ∈ s` the sets `B x : ι x → Set X` with the predicate
`p x : ι x → Prop` form a basis of the filter `𝓝 x`. Then there exists a locally finite covering
`fun i ↦ B (c i) (r i)` of `s` such that all “centers” `c i` belong to `s` and each `r i` satisfies
`p (c i)`.

The notation is inspired by the case `B x r = Metric.ball x r` but the theorem applies to
`nhds_basis_opens` as well. If the covering must be subordinate to some open covering of `s`, then
the user should use a basis obtained by `Filter.HasBasis.restrict_subset` or a similar lemma, see
the proof of `paracompact_of_locallyCompact_sigmaCompact` for an example.

The formalization is based on two [ncatlab](https://ncatlab.org/) proofs:
* [locally compact and sigma compact spaces are paracompact](https://ncatlab.org/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact);
* [open cover of smooth manifold admits locally finite refinement by closed balls](https://ncatlab.org/nlab/show/partition+of+unity#ExistenceOnSmoothManifolds).

See also `refinement_of_locallyCompact_sigmaCompact_of_nhds_basis` for a version of this lemma
dealing with a covering of the whole space.

In most cases (namely, if `B c r ∪ B c r'` is again a set of the form `B c r''`) it is possible
to choose `α = X`. This fact is not yet formalized in `mathlib`. -/
theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [WeaklyLocallyCompactSpace X]
    [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
    {s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)) :
    ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
      (∀ a, c a ∈ s ∧ p (c a) (r a)) ∧
        (s ⊆ ⋃ a, B (c a) (r a)) ∧ LocallyFinite fun a ↦ B (c a) (r a) := by
  classical
    -- For technical reasons we prepend two empty sets to the sequence `CompactExhaustion.choice X`
    set K' : CompactExhaustion X := CompactExhaustion.choice X
    set K : CompactExhaustion X := K'.shiftr.shiftr
    set Kdiff := fun n ↦ K (n + 1) \ interior (K n)
    -- Now we restate some properties of `CompactExhaustion` for `K`/`Kdiff`
    have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) := fun x ↦ by
      simpa only [K'.find_shiftr] using
        diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x)
    have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) :=
      fun n ↦ ((K.isCompact _).diff isOpen_interior).inter_right hs
    -- Next we choose a finite covering `B (c n i) (r n i)` of each
    -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
    have : ∀ (n) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 (x : X) :=
      fun n x ↦ (K.isClosed n).compl_mem_nhds fun hx' ↦ x.2.1.2 <| K.subset_interior_succ _ hx'
    -- Porting note: Commented out `haveI` for now.
    --haveI : ∀ (n) (x : ↑(Kdiff n ∩ s)), Nonempty (ι x) := fun n x ↦ (hB x x.2.2).nonempty
    choose! r hrp hr using fun n (x : ↑(Kdiff (n + 1) ∩ s)) ↦ (hB x x.2.2).mem_iff.1 (this n x)
    have hxr : ∀ (n x) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x := fun n x hx ↦
      (hB x hx.2).mem_of_mem (hrp _ ⟨x, hx⟩)
    choose T hT using fun n ↦ (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n)
    set T' : ∀ n, Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ T n
    -- Finally, we take the union of all these coverings
    refine' ⟨Σn, T' n, fun a ↦ a.2, fun a ↦ r a.1 a.2, _, _, _⟩
    · rintro ⟨n, x, hx⟩
      exact ⟨x.2.2, hrp _ _⟩
    · refine' fun x hx ↦ mem_iUnion.2 _
      

Proof state:
case refine'_2
ι✝ : Type u
X : Type v
Y : Type w
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed s
hB : ∀ x ∈ s, HasBasis (𝓝 x) (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := CompactExhaustion.shiftr (CompactExhaustion.shiftr K')
Kdiff : ℕ → Set X := fun n => K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (CompactExhaustion.find K' x + 1)
Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s)
this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 ↑x
r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x
hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x)
hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (K n)ᶜ
hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n { val := x, property := hx }) ∈ 𝓝 x
T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s)
hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n { val := ↑x, property := (_ : ↑x ∈ Kdiff (n + 1) ∩ s) })
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n)
x : X
hx : x ∈ s
⊢ ∃ i, x ∈ B ((fun a => ↑↑a.snd) i) ((fun a => r a.fst ↑a.snd) i)

Next tactic:
rcases mem_iUnion₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩