awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ (b : ℕ × ι), (fun (ni : ℕ × ι) ↦ D ni.1 ni.2) b) univ

Tactic:
iUnion_eq_univ_iff.2 fun x ↦ ?_

Premises used: Iff.mpr, Set.iUnion_eq_univ_iff
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
refine_1 : ∀ (x : α), ∃ (i : ℕ × ι), x ∈ D i.1 i.2
⊢ Eq.{u_1 + 1} (α := Set α) (⋃ (b : ℕ × ι), (fun (ni : ℕ × ι) ↦ D ni.1 ni.2) b) univ

Tactic:
refine iUnion_eq_univ_iff.2 fun x ↦ ?_

Premises used: Iff.mpr, Set.iUnion_eq_univ_iff
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
refine_1 : ∀ (n : ℕ) (i : ι) (h : x ∈ D n i), ∃ (i : ℕ × ι), x ∈ D i.1 i.2
⊢ ∃ (i : ℕ × ι), x ∈ D i.1 i.2

Tactic:
rcases Dcov x with ⟨n, i, h⟩
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
h : x ∈ D n i
⊢ ∃ (i : ℕ × ι), x ∈ D i.1 i.2

Tactic:
⟨⟨n, i⟩, h⟩

Premises used: Prod.mk, Exists.intro
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
h : x ∈ D n i
⊢ ∃ (i : ℕ × ι), x ∈ D i.1 i.2

Tactic:
exact ⟨⟨n, i⟩, h⟩
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
refine_2 :
  ∀ (n : ℕ) (i : ι) (hn : x ∈ D n i),
    ∃ t ∈ 𝓝 x, Set.Finite.{u_1} (α := ℕ × ι) {i : ℕ × ι | ((fun (ni : ℕ × ι) ↦ D ni.1 ni.2) i ∩ t).Nonempty}
⊢ ∃ t ∈ 𝓝 x, Set.Finite.{u_1} (α := ℕ × ι) {i : ℕ × ι | ((fun (ni : ℕ × ι) ↦ D ni.1 ni.2) i ∩ t).Nonempty}

Tactic:
rcases Dcov x with ⟨n, i, hn⟩
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
hn : x ∈ D n i
⊢ D n i ∈ 𝓝 x

Tactic:
have : D n i ∈ 𝓝 x := IsOpen.mem_nhds (Dopen _ _) hn
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
hn : x ∈ D n i
this : D n i ∈ 𝓝 x
refine_2 :
  ∀ (k : ℕ) (hsub : eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ k) ⊆ D n i),
    ∃ t ∈ 𝓝 x, Set.Finite.{u_1} (α := ℕ × ι) {i : ℕ × ι | ((fun (ni : ℕ × ι) ↦ D ni.1 ni.2) i ∩ t).Nonempty}
⊢ ∃ t ∈ 𝓝 x, Set.Finite.{u_1} (α := ℕ × ι) {i : ℕ × ι | ((fun (ni : ℕ × ι) ↦ D ni.1 ni.2) i ∩ t).Nonempty}

Tactic:
rcases (nhds_basis_uniformity uniformity_basis_edist_inv_two_pow).mem_iff.1 this with
      ⟨k, -, hsub : eball x (2⁻¹ ^ k) ⊆ D n i⟩

Premises used: Filter.HasBasis.mem_iff, uniformity_basis_edist_inv_two_pow, Iff.mp, nhds_basis_uniformity
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
hn : x ∈ D n i
this : D n i ∈ 𝓝 x
k : ℕ
hsub : eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ k) ⊆ D n i
B : Set α := eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + k + 1))
⊢ ∃ t ∈ 𝓝 x, Set.Finite.{u_1} (α := ℕ × ι) {i : ℕ × ι | ((fun (ni : ℕ × ι) ↦ D ni.1 ni.2) i ∩ t).Nonempty}

Tactic:
⟨B, eball_mem_nhds _ (pow_pos _), ?_⟩

Premises used: Metric.eball_mem_nhds, And.intro, Exists.intro
-- Module: Mathlib.Topology.EMetricSpace.Paracompact
Goal:
α : Type u_1
inst : PseudoEMetricSpace α
pow_pos : ∀ (k : ℕ), 0 < HPow.hPow (α := ℝ≥0∞) 2⁻¹ k
hpow_le : ∀ {m n : ℕ}, m ≤ n → HPow.hPow (α := ℝ≥0∞) 2⁻¹ n ≤ HPow.hPow (α := ℝ≥0∞) 2⁻¹ m
h2pow : ∀ (n : ℕ), 2 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + 1) = HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
ι : Type u_1
s : ι → Set α
ho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)
hcov : ∀ (x : α), ∃ (i : ι), x ∈ s i
w : LinearOrder ι
wf : WellFoundedLT ι
ind : (x : α) → ι := fun (x : α) ↦ WellFounded.min (r := fun (x1 x2 : ι) ↦ x1 < x2) ⋯ {i : ι | x ∈ s i} ⋯
mem_ind : ∀ (x : α), x ∈ s (ind x)
notMem_of_lt_ind : ∀ {x : α} {i : ι} (hlt : i < ind x) (hxi : x ∈ s i), False
D : ℕ → ι → Set α :=
  fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
hD :
  D = fun (n : ℕ) ↦
    n.strongRecOn' (P := fun (x : ℕ) ↦ ι → Set α) fun (n : ℕ) (D' : (m : ℕ) → m < n → ι → Set α) (i : ι) ↦
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ (m : ℕ) (H : m < n) (j : ι), x ∉ D' m H j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
Dn :
  ∀ (n : ℕ) (i : ι),
    D n i =
      ⋃ (x : α),
        ⋃ (_ : ind x = i),
          ⋃ (_ : eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i),
            ⋃ (_ : ∀ m < n, ∀ (j : ι), x ∉ D m j), eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ n)
memD :
  ∀ {n : ℕ} {i : ι} {y : α},
    y ∈ D n i ↔
      ∃ (x : α),
        ind x = i ∧
          eball x (3 * HPow.hPow (α := ℝ≥0∞) 2⁻¹ n) ⊆ s i ∧
            (∀ m < n, ∀ (j : ι), x ∉ D m j) ∧ edist y x < HPow.hPow (α := ℝ≥0∞) 2⁻¹ n
Dcov : ∀ (x : α), ∃ (n : ℕ) (i : ι), x ∈ D n i
Dopen : ∀ (n : ℕ) (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (D n i)
HDS : ∀ (n : ℕ) (i : ι), D n i ⊆ s i
x : α
n : ℕ
i : ι
hn : x ∈ D n i
this : D n i ∈ 𝓝 x
k : ℕ
hsub : eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ k) ⊆ D n i
B : Set α := eball x (HPow.hPow (α := ℝ≥0∞) 2⁻¹ (n + k + 1))
refine_2 : Set.Finite.{u_1} (α := ℕ × ι) {i : ℕ × ι | ((fun (ni : ℕ × ι) ↦ D ni.1 ni.2) i ∩ B).Nonempty}
⊢ ∃ t ∈ 𝓝 x, Set.Finite.{u_1} (α := ℕ × ι) {i : ℕ × ι | ((fun (ni : ℕ × ι) ↦ D ni.1 ni.2) i ∩ t).Nonempty}

Tactic:
refine ⟨B, eball_mem_nhds _ (pow_pos _), ?_⟩