Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
inst✝³ : TopologicalSpace X
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[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find 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 ⟨x, 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 ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ ↑(T n)
⊢ ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a ↦ B (c a) (r a)

Tactic:
refine ⟨Σ n, T' n, fun a ↦ a.2, fun a ↦ r a.1 a.2, ?_, ?_, ?_⟩

Goal state after:
case refine_1
X : Type v
inst✝³ : TopologicalSpace X
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[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find 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 ⟨x, 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 ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ ↑(T n)
⊢ ∀ (a : (n : ℕ) × ↑(T' n)), (fun a ↦ ↑↑a.snd) a ∈ s ∧ p ((fun a ↦ ↑↑a.snd) a) ((fun a ↦ r a.fst ↑a.snd) a)

case refine_2
X : Type v
inst✝³ : TopologicalSpace X
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[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find 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 ⟨x, 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 ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ ↑(T n)
⊢ s ⊆ ⋃ a, B ((fun a ↦ ↑↑a.snd) a) ((fun a ↦ r a.fst ↑a.snd) a)

case refine_3
X : Type v
inst✝³ : TopologicalSpace X
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[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find 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 ⟨x, 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 ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ ↑(T n)
⊢ LocallyFinite fun a ↦ B ((fun a ↦ ↑↑a.snd) a) ((fun a ↦ r a.fst ↑a.snd) a)