l3lab/ntp-mathlib huggingface dataset:
/-- In general it suffices that all but finitely many of the spaces are compact,
  but that's not straightforward to state and use. -/
instance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    

Proof state:
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : Finite ι
t : (i : ι) → X i
n : Set ((i : ι) → X i)
hn : n ∈ 𝓝 t
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
rw [nhds_pi, Filter.mem_pi] at hn
/-- In general it suffices that all but finitely many of the spaces are compact,
  but that's not straightforward to state and use. -/
instance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    

Proof state:
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : Finite ι
t : (i : ι) → X i
n : Set ((i : ι) → X i)
hn : ∃ I, Set.Finite I ∧ ∃ t_1, (∀ (i : ι), t_1 i ∈ 𝓝 (t i)) ∧ Set.pi I t_1 ⊆ n
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
obtain ⟨s, -, n', hn', hsub⟩ := hn
/-- In general it suffices that all but finitely many of the spaces are compact,
  but that's not straightforward to state and use. -/
instance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, -, n', hn', hsub⟩ := hn
    

Proof state:
case intro.intro.intro.intro
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : Finite ι
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
/-- In general it suffices that all but finitely many of the spaces are compact,
  but that's not straightforward to state and use. -/
instance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, -, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    

Proof state:
case intro.intro.intro.intro
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : Finite ι
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
refine' ⟨(Set.univ : Set ι).pi n'', _, subset_trans (fun _ h => _) hsub, isCompact_univ_pi hc⟩
/-- In general it suffices that all but finitely many of the spaces are compact,
  but that's not straightforward to state and use. -/
instance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, -, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨(Set.univ : Set ι).pi n'', _, subset_trans (fun _ h => _) hsub, isCompact_univ_pi hc⟩
    · 

Proof state:
case intro.intro.intro.intro.refine'_1
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : Finite ι
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
⊢ Set.pi univ n'' ∈ 𝓝 t

Next tactic:
exact (set_pi_mem_nhds_iff (@Set.finite_univ ι _) _).mpr fun i _ => hn'' i
/-- In general it suffices that all but finitely many of the spaces are compact,
  but that's not straightforward to state and use. -/
instance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, -, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨(Set.univ : Set ι).pi n'', _, subset_trans (fun _ h => _) hsub, isCompact_univ_pi hc⟩
    · exact (set_pi_mem_nhds_iff (@Set.finite_univ ι _) _).mpr fun i _ => hn'' i
    · 

Proof state:
case intro.intro.intro.intro.refine'_2
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : Finite ι
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
x✝ : (i : ι) → X i
h : x✝ ∈ Set.pi univ n''
⊢ x✝ ∈ Set.pi s n'

Next tactic:
exact fun i _ => hsub' i (h i trivial)
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    

Proof state:
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
hn : n ∈ 𝓝 t
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
rw [nhds_pi, Filter.mem_pi] at hn
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    

Proof state:
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
hn : ∃ I, Set.Finite I ∧ ∃ t_1, (∀ (i : ι), t_1 i ∈ 𝓝 (t i)) ∧ Set.pi I t_1 ⊆ n
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
obtain ⟨s, hs, n', hn', hsub⟩ := hn
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    

Proof state:
case intro.intro.intro.intro
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    

Proof state:
case intro.intro.intro.intro
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
⊢ ∃ s ∈ 𝓝 t, s ⊆ n ∧ IsCompact s

Next tactic:
refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · 

Proof state:
case intro.intro.intro.intro.refine'_1
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
⊢ Set.pi s n'' ∈ 𝓝 t

Next tactic:
exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · 

Proof state:
case intro.intro.intro.intro.refine'_2
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
x✝ : (i : ι) → X i
⊢ x✝ ∈ Set.pi s n'' → x✝ ∈ Set.pi s n'

Next tactic:
exact forall₂_imp fun i _ hi' => hsub' i hi'
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · exact forall₂_imp fun i _ hi' => hsub' i hi'
    · 

Proof state:
case intro.intro.intro.intro.refine'_3
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
⊢ IsCompact (Set.pi s n'')

Next tactic:
rw [← Set.univ_pi_ite]
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · exact forall₂_imp fun i _ hi' => hsub' i hi'
    · rw [← Set.univ_pi_ite]
      

Proof state:
case intro.intro.intro.intro.refine'_3
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
⊢ IsCompact (Set.pi univ fun i => if i ∈ s then n'' i else univ)

Next tactic:
refine' isCompact_univ_pi fun i => _
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · exact forall₂_imp fun i _ hi' => hsub' i hi'
    · rw [← Set.univ_pi_ite]
      refine' isCompact_univ_pi fun i => _
      

Proof state:
case intro.intro.intro.intro.refine'_3
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
i : ι
⊢ IsCompact (if i ∈ s then n'' i else univ)

Next tactic:
by_cases h : i ∈ s
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · exact forall₂_imp fun i _ hi' => hsub' i hi'
    · rw [← Set.univ_pi_ite]
      refine' isCompact_univ_pi fun i => _
      by_cases h : i ∈ s
      · 

Proof state:
case pos
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
i : ι
h : i ∈ s
⊢ IsCompact (if i ∈ s then n'' i else univ)

Next tactic:
rw [if_pos h]
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · exact forall₂_imp fun i _ hi' => hsub' i hi'
    · rw [← Set.univ_pi_ite]
      refine' isCompact_univ_pi fun i => _
      by_cases h : i ∈ s
      · rw [if_pos h]
        

Proof state:
case pos
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
i : ι
h : i ∈ s
⊢ IsCompact (n'' i)

Next tactic:
exact hc i
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · exact forall₂_imp fun i _ hi' => hsub' i hi'
    · rw [← Set.univ_pi_ite]
      refine' isCompact_univ_pi fun i => _
      by_cases h : i ∈ s
      · rw [if_pos h]
        exact hc i
      · 

Proof state:
case neg
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
i : ι
h : i ∉ s
⊢ IsCompact (if i ∈ s then n'' i else univ)

Next tactic:
rw [if_neg h]
/-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=
  ⟨fun t n hn => by
    rw [nhds_pi, Filter.mem_pi] at hn
    obtain ⟨s, hs, n', hn', hsub⟩ := hn
    choose n'' hn'' hsub' hc using fun i =>
      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)
    refine' ⟨s.pi n'', _, subset_trans (fun _ => _) hsub, _⟩
    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i
    · exact forall₂_imp fun i _ hi' => hsub' i hi'
    · rw [← Set.univ_pi_ite]
      refine' isCompact_univ_pi fun i => _
      by_cases h : i ∈ s
      · rw [if_pos h]
        exact hc i
      · rw [if_neg h]
        

Proof state:
case neg
X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s✝ t✝ : Set X✝
X : ι → Type u_4
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
t : (i : ι) → X i
n : Set ((i : ι) → X i)
s : Set ι
hs : Set.Finite s
n' : (i : ι) → Set (X i)
hn' : ∀ (i : ι), n' i ∈ 𝓝 (t i)
hsub : Set.pi s n' ⊆ n
n'' : (i : ι) → Set (X i)
hn'' : ∀ (i : ι), n'' i ∈ 𝓝 (t i)
hsub' : ∀ (i : ι), n'' i ⊆ n' i
hc : ∀ (i : ι), IsCompact (n'' i)
i : ι
h : i ∉ s
⊢ IsCompact univ

Next tactic:
exact CompactSpace.isCompact_univ
/-- A reformulation of the definition of locally compact space: In a locally compact space,
  every open set containing `x` has a compact subset containing `x` in its interior. -/
theorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)
    (hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U := by
  

Proof state:
X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : LocallyCompactSpace X
x : X
U : Set X
hU : IsOpen U
hx : x ∈ U
⊢ ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U

Next tactic:
rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩
/-- A reformulation of the definition of locally compact space: In a locally compact space,
  every open set containing `x` has a compact subset containing `x` in its interior. -/
theorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)
    (hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U := by
  rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩
  

Proof state:
case intro.intro.intro
X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : LocallyCompactSpace X
x : X
U : Set X
hU : IsOpen U
hx : x ∈ U
K : Set X
h1K : K ∈ 𝓝 x
h2K : K ⊆ U
h3K : IsCompact K
⊢ ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U

Next tactic:
exact ⟨K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K⟩
/-- In a locally compact space, for every containment `K ⊆ U` of a compact set `K` in an open
  set `U`, there is a compact neighborhood `L` such that `K ⊆ L ⊆ U`: equivalently, there is a
  compact `L` such that `K ⊆ interior L` and `L ⊆ U`.
  See also `exists_compact_closed_between`, in which one guarantees additionally that `L` is closed
  if the space is regular. -/
theorem exists_compact_between [hX : LocallyCompactSpace X] {K U : Set X} (hK : IsCompact K)
    (hU : IsOpen U) (h_KU : K ⊆ U) : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U := by
  

Proof state:
X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
hX : LocallyCompactSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen U
h_KU : K ⊆ U
⊢ ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U

Next tactic:
choose V hVc hxV hKV using fun x : K => exists_compact_subset hU (h_KU x.2)
/-- In a locally compact space, for every containment `K ⊆ U` of a compact set `K` in an open
  set `U`, there is a compact neighborhood `L` such that `K ⊆ L ⊆ U`: equivalently, there is a
  compact `L` such that `K ⊆ interior L` and `L ⊆ U`.
  See also `exists_compact_closed_between`, in which one guarantees additionally that `L` is closed
  if the space is regular. -/
theorem exists_compact_between [hX : LocallyCompactSpace X] {K U : Set X} (hK : IsCompact K)
    (hU : IsOpen U) (h_KU : K ⊆ U) : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U := by
  choose V hVc hxV hKV using fun x : K => exists_compact_subset hU (h_KU x.2)
  

Proof state:
X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
hX : LocallyCompactSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen U
h_KU : K ⊆ U
V : ↑K → Set X
hVc : ∀ (x : ↑K), IsCompact (V x)
hxV : ∀ (x : ↑K), ↑x ∈ interior (V x)
hKV : ∀ (x : ↑K), V x ⊆ U
⊢ ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U

Next tactic:
have : K ⊆ ⋃ x, interior (V x) := fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, hxV _⟩