l3lab/ntp-mathlib huggingface dataset:
instance : CompleteSpace (CauchyFilter α) :=
  completeSpace_extension uniformInducing_pureCauchy denseRange_pureCauchy fun f hf =>
    let f' : CauchyFilter α := ⟨f, hf⟩
    have : map pureCauchy f ≤ (𝓤 <| CauchyFilter α).lift' (preimage (Prod.mk f')) :=
      le_lift'.2 fun s hs =>
        let ⟨t, ht₁, (ht₂ : gen t ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
        let ⟨t', ht', (h : t' ×ˢ t' ⊆ t)⟩ := mem_prod_same_iff.mp (hf.right ht₁)
        have : t' ⊆ { y : α | (f', pureCauchy y) ∈ gen t } := fun x hx =>
          (f ×ˢ pure x).sets_of_superset (prod_mem_prod ht' hx) h
        f.sets_of_superset ht' <| Subset.trans this (preimage_mono ht₂)
    ⟨f', by 

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : Filter α
hf : Cauchy f
f' : CauchyFilter α := { val := f, property := hf }
this : map pureCauchy f ≤ Filter.lift' (𝓤 (CauchyFilter α)) (preimage (Prod.mk f'))
⊢ map pureCauchy f ≤ 𝓝 f'

Next tactic:
simp [nhds_eq_uniformity]
instance : CompleteSpace (CauchyFilter α) :=
  completeSpace_extension uniformInducing_pureCauchy denseRange_pureCauchy fun f hf =>
    let f' : CauchyFilter α := ⟨f, hf⟩
    have : map pureCauchy f ≤ (𝓤 <| CauchyFilter α).lift' (preimage (Prod.mk f')) :=
      le_lift'.2 fun s hs =>
        let ⟨t, ht₁, (ht₂ : gen t ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
        let ⟨t', ht', (h : t' ×ˢ t' ⊆ t)⟩ := mem_prod_same_iff.mp (hf.right ht₁)
        have : t' ⊆ { y : α | (f', pureCauchy y) ∈ gen t } := fun x hx =>
          (f ×ˢ pure x).sets_of_superset (prod_mem_prod ht' hx) h
        f.sets_of_superset ht' <| Subset.trans this (preimage_mono ht₂)
    ⟨f', by simp [nhds_eq_uniformity]; 

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : Filter α
hf : Cauchy f
f' : CauchyFilter α := { val := f, property := hf }
this : map pureCauchy f ≤ Filter.lift' (𝓤 (CauchyFilter α)) (preimage (Prod.mk f'))
⊢ map pureCauchy f ≤ Filter.lift' (𝓤 (CauchyFilter α)) (UniformSpace.ball { val := f, property := hf })

Next tactic:
assumption
theorem extend_pureCauchy {f : α → β} (hf : UniformContinuous f) (a : α) :
    extend f (pureCauchy a) = f a := by
  

Proof state:
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : SeparatedSpace β
f : α → β
hf : UniformContinuous f
a : α
⊢ extend f (pureCauchy a) = f a

Next tactic:
rw [extend, if_pos hf]
theorem extend_pureCauchy {f : α → β} (hf : UniformContinuous f) (a : α) :
    extend f (pureCauchy a) = f a := by
  rw [extend, if_pos hf]
  

Proof state:
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : SeparatedSpace β
f : α → β
hf : UniformContinuous f
a : α
⊢ DenseInducing.extend (_ : DenseInducing pureCauchy) f (pureCauchy a) = f a

Next tactic:
exact uniformly_extend_of_ind uniformInducing_pureCauchy denseRange_pureCauchy hf _
theorem uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) := by
  

Proof state:
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace β
f : α → β
⊢ UniformContinuous (extend f)

Next tactic:
by_cases hf : UniformContinuous f
theorem uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) := by
  by_cases hf : UniformContinuous f
  · 

Proof state:
case pos
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace β
f : α → β
hf : UniformContinuous f
⊢ UniformContinuous (extend f)

Next tactic:
rw [extend, if_pos hf]
theorem uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) := by
  by_cases hf : UniformContinuous f
  · rw [extend, if_pos hf]
    

Proof state:
case pos
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace β
f : α → β
hf : UniformContinuous f
⊢ UniformContinuous (DenseInducing.extend (_ : DenseInducing pureCauchy) f)

Next tactic:
exact uniformContinuous_uniformly_extend uniformInducing_pureCauchy denseRange_pureCauchy hf
theorem uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) := by
  by_cases hf : UniformContinuous f
  · rw [extend, if_pos hf]
    exact uniformContinuous_uniformly_extend uniformInducing_pureCauchy denseRange_pureCauchy hf
  · 

Proof state:
case neg
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace β
f : α → β
hf : ¬UniformContinuous f
⊢ UniformContinuous (extend f)

Next tactic:
rw [extend, if_neg hf]
theorem uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) := by
  by_cases hf : UniformContinuous f
  · rw [extend, if_pos hf]
    exact uniformContinuous_uniformly_extend uniformInducing_pureCauchy denseRange_pureCauchy hf
  · rw [extend, if_neg hf]
    

Proof state:
case neg
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace β
f : α → β
hf : ¬UniformContinuous f
⊢ UniformContinuous fun x => f (Nonempty.some (_ : Nonempty α))

Next tactic:
exact uniformContinuous_of_const fun a _b => by congr
theorem uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) := by
  by_cases hf : UniformContinuous f
  · rw [extend, if_pos hf]
    exact uniformContinuous_uniformly_extend uniformInducing_pureCauchy denseRange_pureCauchy hf
  · rw [extend, if_neg hf]
    exact uniformContinuous_of_const fun a _b => by 

Proof state:
α : Type u
inst✝³ : UniformSpace α
β : Type v
γ : Type w
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace β
f : α → β
hf : ¬UniformContinuous f
a _b : CauchyFilter α
⊢ f (Nonempty.some (_ : Nonempty α)) = f (Nonempty.some (_ : Nonempty α))

Next tactic:
congr
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  

Proof state:
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
⊢ lim ↑f = lim ↑g ↔ (f, g) ∈ 𝓢 (CauchyFilter α)

Next tactic:
constructor
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · 

Proof state:
case mp
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
⊢ lim ↑f = lim ↑g → (f, g) ∈ 𝓢 (CauchyFilter α)

Next tactic:
intro e s hs
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    

Proof state:
case mp
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
⊢ (f, g) ∈ s

Next tactic:
rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    

Proof state:
case mp.intro.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
t : Set (α × α)
tu : t ∈ 𝓤 α
ts : ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s
⊢ (f, g) ∈ s

Next tactic:
apply ts
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    

Proof state:
case mp.intro.intro.a
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
t : Set (α × α)
tu : t ∈ 𝓤 α
ts : ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s
⊢ t ∈ ↑f ×ˢ ↑g

Next tactic:
rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    

Proof state:
case mp.intro.intro.a.intro.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
t : Set (α × α)
tu : t ∈ 𝓤 α
ts : ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s
d : Set (α × α)
du : d ∈ 𝓤 α
dt : d ○ d ⊆ t
⊢ t ∈ ↑f ×ˢ ↑g

Next tactic:
refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    

Proof state:
case mp.intro.intro.a.intro.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
t : Set (α × α)
tu : t ∈ 𝓤 α
ts : ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s
d : Set (α × α)
du : d ∈ 𝓤 α
dt : d ○ d ⊆ t
x : α × α
h : x ∈ {x | (x, lim ↑f) ∈ d} ×ˢ {y | (lim ↑g, y) ∈ d}
⊢ x ∈ t

Next tactic:
cases' x with a b
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    

Proof state:
case mp.intro.intro.a.intro.intro.mk
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
t : Set (α × α)
tu : t ∈ 𝓤 α
ts : ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s
d : Set (α × α)
du : d ∈ 𝓤 α
dt : d ○ d ⊆ t
a b : α
h : (a, b) ∈ {x | (x, lim ↑f) ∈ d} ×ˢ {y | (lim ↑g, y) ∈ d}
⊢ (a, b) ∈ t

Next tactic:
cases' h with h₁ h₂
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    

Proof state:
case mp.intro.intro.a.intro.intro.mk.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
t : Set (α × α)
tu : t ∈ 𝓤 α
ts : ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s
d : Set (α × α)
du : d ∈ 𝓤 α
dt : d ○ d ⊆ t
a b : α
h₁ : (a, b).1 ∈ {x | (x, lim ↑f) ∈ d}
h₂ : (a, b).2 ∈ {y | (lim ↑g, y) ∈ d}
⊢ (a, b) ∈ t

Next tactic:
rw [← e] at h₂
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    

Proof state:
case mp.intro.intro.a.intro.intro.mk.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
e : lim ↑f = lim ↑g
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ (𝓤 (CauchyFilter α)).sets
t : Set (α × α)
tu : t ∈ 𝓤 α
ts : ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s
d : Set (α × α)
du : d ∈ 𝓤 α
dt : d ○ d ⊆ t
a b : α
h₁ : (a, b).1 ∈ {x | (x, lim ↑f) ∈ d}
h₂ : (a, b).2 ∈ {y | (lim ↑f, y) ∈ d}
⊢ (a, b) ∈ t

Next tactic:
exact dt ⟨_, h₁, h₂⟩
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · 

Proof state:
case mpr
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
⊢ (f, g) ∈ 𝓢 (CauchyFilter α) → lim ↑f = lim ↑g

Next tactic:
intro H
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · intro H
    

Proof state:
case mpr
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
H : (f, g) ∈ 𝓢 (CauchyFilter α)
⊢ lim ↑f = lim ↑g

Next tactic:
refine' separated_def.1 (by infer_instance) _ _ fun t tu => _
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · intro H
    refine' separated_def.1 (by 

Proof state:
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
H : (f, g) ∈ 𝓢 (CauchyFilter α)
⊢ SeparatedSpace α

Next tactic:
infer_instance
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · intro H
    refine' separated_def.1 (by infer_instance) _ _ fun t tu => _
    

Proof state:
case mpr
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
H : (f, g) ∈ 𝓢 (CauchyFilter α)
t : Set (α × α)
tu : t ∈ 𝓤 α
⊢ (lim ↑f, lim ↑g) ∈ t

Next tactic:
rcases mem_uniformity_isClosed tu with ⟨d, du, dc, dt⟩
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · intro H
    refine' separated_def.1 (by infer_instance) _ _ fun t tu => _
    rcases mem_uniformity_isClosed tu with ⟨d, du, dc, dt⟩
    

Proof state:
case mpr.intro.intro.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f g : CauchyFilter α
H : (f, g) ∈ 𝓢 (CauchyFilter α)
t : Set (α × α)
tu : t ∈ 𝓤 α
d : Set (α × α)
du : d ∈ 𝓤 α
dc : IsClosed d
dt : d ⊆ t
⊢ (lim ↑f, lim ↑g) ∈ t

Next tactic:
refine'
      H { p | (lim p.1.1, lim p.2.1) ∈ t } (CauchyFilter.mem_uniformity'.2 ⟨d, du, fun f g h => _⟩)
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · intro H
    refine' separated_def.1 (by infer_instance) _ _ fun t tu => _
    rcases mem_uniformity_isClosed tu with ⟨d, du, dc, dt⟩
    refine'
      H { p | (lim p.1.1, lim p.2.1) ∈ t } (CauchyFilter.mem_uniformity'.2 ⟨d, du, fun f g h => _⟩)
    

Proof state:
case mpr.intro.intro.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f✝ g✝ : CauchyFilter α
H : (f✝, g✝) ∈ 𝓢 (CauchyFilter α)
t : Set (α × α)
tu : t ∈ 𝓤 α
d : Set (α × α)
du : d ∈ 𝓤 α
dc : IsClosed d
dt : d ⊆ t
f g : CauchyFilter α
h : d ∈ ↑f ×ˢ ↑g
⊢ (f, g) ∈ {p | (lim ↑p.1, lim ↑p.2) ∈ t}

Next tactic:
rcases mem_prod_iff.1 h with ⟨x, xf, y, yg, h⟩
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · intro H
    refine' separated_def.1 (by infer_instance) _ _ fun t tu => _
    rcases mem_uniformity_isClosed tu with ⟨d, du, dc, dt⟩
    refine'
      H { p | (lim p.1.1, lim p.2.1) ∈ t } (CauchyFilter.mem_uniformity'.2 ⟨d, du, fun f g h => _⟩)
    rcases mem_prod_iff.1 h with ⟨x, xf, y, yg, h⟩
    

Proof state:
case mpr.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f✝ g✝ : CauchyFilter α
H : (f✝, g✝) ∈ 𝓢 (CauchyFilter α)
t : Set (α × α)
tu : t ∈ 𝓤 α
d : Set (α × α)
du : d ∈ 𝓤 α
dc : IsClosed d
dt : d ⊆ t
f g : CauchyFilter α
h✝ : d ∈ ↑f ×ˢ ↑g
x : Set α
xf : x ∈ ↑f
y : Set α
yg : y ∈ ↑g
h : x ×ˢ y ⊆ d
⊢ (f, g) ∈ {p | (lim ↑p.1, lim ↑p.2) ∈ t}

Next tactic:
have limc : ∀ (f : CauchyFilter α), ∀ x ∈ f.1, lim f.1 ∈ closure x := by
      intro f x xf
      rw [closure_eq_cluster_pts]
      exact f.2.1.mono (le_inf f.2.le_nhds_lim (le_principal_iff.2 xf))
theorem cauchyFilter_eq {α : Type*} [Inhabited α] [UniformSpace α] [CompleteSpace α]
    [SeparatedSpace α] {f g : CauchyFilter α} :
    lim f.1 = lim g.1 ↔ (f, g) ∈ separationRel (CauchyFilter α) := by
  constructor
  · intro e s hs
    rcases CauchyFilter.mem_uniformity'.1 hs with ⟨t, tu, ts⟩
    apply ts
    rcases comp_mem_uniformity_sets tu with ⟨d, du, dt⟩
    refine'
      mem_prod_iff.2
        ⟨_, f.2.le_nhds_lim (mem_nhds_right (lim f.1) du), _,
          g.2.le_nhds_lim (mem_nhds_left (lim g.1) du), fun x h => _⟩
    cases' x with a b
    cases' h with h₁ h₂
    rw [← e] at h₂
    exact dt ⟨_, h₁, h₂⟩
  · intro H
    refine' separated_def.1 (by infer_instance) _ _ fun t tu => _
    rcases mem_uniformity_isClosed tu with ⟨d, du, dc, dt⟩
    refine'
      H { p | (lim p.1.1, lim p.2.1) ∈ t } (CauchyFilter.mem_uniformity'.2 ⟨d, du, fun f g h => _⟩)
    rcases mem_prod_iff.1 h with ⟨x, xf, y, yg, h⟩
    have limc : ∀ (f : CauchyFilter α), ∀ x ∈ f.1, lim f.1 ∈ closure x := by
      

Proof state:
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f✝ g✝ : CauchyFilter α
H : (f✝, g✝) ∈ 𝓢 (CauchyFilter α)
t : Set (α × α)
tu : t ∈ 𝓤 α
d : Set (α × α)
du : d ∈ 𝓤 α
dc : IsClosed d
dt : d ⊆ t
f g : CauchyFilter α
h✝ : d ∈ ↑f ×ˢ ↑g
x : Set α
xf : x ∈ ↑f
y : Set α
yg : y ∈ ↑g
h : x ×ˢ y ⊆ d
⊢ ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x

Next tactic:
intro f x xf