l3lab/ntp-mathlib huggingface dataset:
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : s ∈ l
this✝ : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
this : ∀ (n : ℕ), W n ∩ s ×ˢ s ∈ l ×ˢ l
⊢ ∀ (n : ℕ), ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s

Next tactic:
simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this✝ : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
this : ∀ (n : ℕ), ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s
⊢ ∃ t, Antitone t ∧ (∀ (n : ℕ), t n ∈ l) ∧ (∀ (n : ℕ), t n ×ˢ t n ⊆ W n) ∧ ∀ (n : ℕ), t n ⊆ s

Next tactic:
choose t htl htW hts using this
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
⊢ ∃ t, Antitone t ∧ (∀ (n : ℕ), t n ∈ l) ∧ (∀ (n : ℕ), t n ×ˢ t n ⊆ W n) ∧ ∀ (n : ℕ), t n ⊆ s

Next tactic:
have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by 

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
n : ℕ
⊢ ⋂ k, ⋂ (_ : k ≤ n), t k ⊆ t n

Next tactic:
apply iInter₂_subset
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; 

Proof state:
case j
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
n : ℕ
⊢ n ≤ n

Next tactic:
rfl
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this✝ : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
this : ∀ (n : ℕ), ⋂ k, ⋂ (_ : k ≤ n), t k ⊆ t n
⊢ ∃ t, Antitone t ∧ (∀ (n : ℕ), t n ∈ l) ∧ (∀ (n : ℕ), t n ×ˢ t n ⊆ W n) ∧ ∀ (n : ℕ), t n ⊆ s

Next tactic:
exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
  

Proof state:
case intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
choose u hu using fun n => Filter.nonempty_of_mem (htl n)
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
  choose u hu using fun n => Filter.nonempty_of_mem (htl n)
  

Proof state:
case intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
have huc : CauchySeq u := hV.toHasBasis.cauchySeq_iff.2 fun N _ =>
      ⟨N, fun m hm n hn => hWV' _ <| @htW N (_, _) ⟨ht_anti hm (hu _), ht_anti hn (hu _)⟩⟩
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
  choose u hu using fun n => Filter.nonempty_of_mem (htl n)
  have huc : CauchySeq u := hV.toHasBasis.cauchySeq_iff.2 fun N _ =>
      ⟨N, fun m hm n hn => hWV' _ <| @htW N (_, _) ⟨ht_anti hm (hu _), ht_anti hn (hu _)⟩⟩
  

Proof state:
case intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
rcases hs.exists_tendsto (fun n => hts n (hu n)) huc with ⟨x, hxs, hx⟩
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
  choose u hu using fun n => Filter.nonempty_of_mem (htl n)
  have huc : CauchySeq u := hV.toHasBasis.cauchySeq_iff.2 fun N _ =>
      ⟨N, fun m hm n hn => hWV' _ <| @htW N (_, _) ⟨ht_anti hm (hu _), ht_anti hn (hu _)⟩⟩
  rcases hs.exists_tendsto (fun n => hts n (hu n)) huc with ⟨x, hxs, hx⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
refine ⟨x, hxs, (nhds_basis_uniformity' hV.toHasBasis).ge_iff.2 fun N _ => ?_⟩
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
  choose u hu using fun n => Filter.nonempty_of_mem (htl n)
  have huc : CauchySeq u := hV.toHasBasis.cauchySeq_iff.2 fun N _ =>
      ⟨N, fun m hm n hn => hWV' _ <| @htW N (_, _) ⟨ht_anti hm (hu _), ht_anti hn (hu _)⟩⟩
  rcases hs.exists_tendsto (fun n => hts n (hu n)) huc with ⟨x, hxs, hx⟩
  refine ⟨x, hxs, (nhds_basis_uniformity' hV.toHasBasis).ge_iff.2 fun N _ => ?_⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
N : ℕ
x✝ : True
⊢ ball x (V N) ∈ l

Next tactic:
obtain ⟨n, hNn, hn⟩ : ∃ n, N ≤ n ∧ u n ∈ ball x (W N)
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
  choose u hu using fun n => Filter.nonempty_of_mem (htl n)
  have huc : CauchySeq u := hV.toHasBasis.cauchySeq_iff.2 fun N _ =>
      ⟨N, fun m hm n hn => hWV' _ <| @htW N (_, _) ⟨ht_anti hm (hu _), ht_anti hn (hu _)⟩⟩
  rcases hs.exists_tendsto (fun n => hts n (hu n)) huc with ⟨x, hxs, hx⟩
  refine ⟨x, hxs, (nhds_basis_uniformity' hV.toHasBasis).ge_iff.2 fun N _ => ?_⟩
  obtain ⟨n, hNn, hn⟩ : ∃ n, N ≤ n ∧ u n ∈ ball x (W N)
  · 

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
t : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
N : ℕ
x✝ : True
⊢ ∃ n, N ≤ n ∧ u n ∈ ball x (W N)

Next tactic:
exact ((eventually_ge_atTop N).and (hx <| ball_mem_nhds x (hW N))).exists