l3lab/ntp-mathlib huggingface dataset:
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
      intro f x xf
      

Proof state:
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f✝¹ g✝ : CauchyFilter α
H : (f✝¹, 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 : Set α
xf : x ∈ ↑f
⊢ lim ↑f ∈ closure x

Next tactic:
rw [closure_eq_cluster_pts]
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
      intro f x xf
      rw [closure_eq_cluster_pts]
      

Proof state:
α : Type u_1
inst✝³ : Inhabited α
inst✝² : UniformSpace α
inst✝¹ : CompleteSpace α
inst✝ : SeparatedSpace α
f✝¹ g✝ : CauchyFilter α
H : (f✝¹, 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 : Set α
xf : x ∈ ↑f
⊢ lim ↑f ∈ {a | ClusterPt a (𝓟 x)}

Next tactic:
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
      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))
    

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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
⊢ (f, g) ∈ {p | (lim ↑p.1, lim ↑p.2) ∈ t}

Next tactic:
have := dc.closure_subset_iff.2 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⟩
    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))
    have := dc.closure_subset_iff.2 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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure (x ×ˢ y) ⊆ d
⊢ (f, g) ∈ {p | (lim ↑p.1, lim ↑p.2) ∈ t}

Next tactic:
rw [closure_prod_eq] at this
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
      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))
    have := dc.closure_subset_iff.2 h
    rw [closure_prod_eq] at this
    

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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure x ×ˢ closure y ⊆ d
⊢ (f, g) ∈ {p | (lim ↑p.1, lim ↑p.2) ∈ t}

Next tactic:
refine' dt (this ⟨_, _⟩)
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
      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))
    have := dc.closure_subset_iff.2 h
    rw [closure_prod_eq] at this
    refine' dt (this ⟨_, _⟩) <;> 

Proof state:
case mpr.intro.intro.intro.intro.intro.intro.intro.refine'_1
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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure x ×ˢ closure y ⊆ d
⊢ (lim ↑(f, g).1, lim ↑(f, g).2).1 ∈ closure x

Next tactic:
dsimp
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
      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))
    have := dc.closure_subset_iff.2 h
    rw [closure_prod_eq] at this
    refine' dt (this ⟨_, _⟩) <;> 

Proof state:
case mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2
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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure x ×ˢ closure y ⊆ d
⊢ (lim ↑(f, g).1, lim ↑(f, g).2).2 ∈ closure y

Next tactic:
dsimp
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
      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))
    have := dc.closure_subset_iff.2 h
    rw [closure_prod_eq] at this
    refine' dt (this ⟨_, _⟩) <;> dsimp <;> 

Proof state:
case mpr.intro.intro.intro.intro.intro.intro.intro.refine'_1
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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure x ×ˢ closure y ⊆ d
⊢ lim ↑f ∈ closure x

Next tactic:
apply limc
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
      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))
    have := dc.closure_subset_iff.2 h
    rw [closure_prod_eq] at this
    refine' dt (this ⟨_, _⟩) <;> dsimp <;> 

Proof state:
case mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2
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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure x ×ˢ closure y ⊆ d
⊢ lim ↑g ∈ closure y

Next tactic:
apply limc
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
      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))
    have := dc.closure_subset_iff.2 h
    rw [closure_prod_eq] at this
    refine' dt (this ⟨_, _⟩) <;> dsimp <;> apply limc <;> 

Proof state:
case mpr.intro.intro.intro.intro.intro.intro.intro.refine'_1.a
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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure x ×ˢ closure y ⊆ d
⊢ x ∈ ↑f

Next tactic:
assumption
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
      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))
    have := dc.closure_subset_iff.2 h
    rw [closure_prod_eq] at this
    refine' dt (this ⟨_, _⟩) <;> dsimp <;> apply limc <;> 

Proof state:
case mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2.a
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
limc : ∀ (f : CauchyFilter α), ∀ x ∈ ↑f, lim ↑f ∈ closure x
this : closure x ×ˢ closure y ⊆ d
⊢ y ∈ ↑g

Next tactic:
assumption
theorem separated_pureCauchy_injective {α : Type*} [UniformSpace α] [s : SeparatedSpace α] :
    @Function.Injective α (Quotient (UniformSpace.separationSetoid (CauchyFilter α)))
      fun a : α => ⟦pureCauchy a⟧
  | a, b, h => by
    

Proof state:
α : Type u_1
inst✝ : UniformSpace α
s : SeparatedSpace α
a b : α
h : (fun a => ⟦pureCauchy a⟧) a = (fun a => ⟦pureCauchy a⟧) b
⊢ a = b

Next tactic:
refine separated_def.1 s _ _ (fun s hs => ?_)
theorem separated_pureCauchy_injective {α : Type*} [UniformSpace α] [s : SeparatedSpace α] :
    @Function.Injective α (Quotient (UniformSpace.separationSetoid (CauchyFilter α)))
      fun a : α => ⟦pureCauchy a⟧
  | a, b, h => by
    refine separated_def.1 s _ _ (fun s hs => ?_)
    

Proof state:
α : Type u_1
inst✝ : UniformSpace α
s✝ : SeparatedSpace α
a b : α
h : (fun a => ⟦pureCauchy a⟧) a = (fun a => ⟦pureCauchy a⟧) b
s : Set (α × α)
hs : s ∈ 𝓤 α
⊢ (a, b) ∈ s

Next tactic:
rw [← (@uniformEmbedding_pureCauchy α _).comap_uniformity, Filter.mem_comap] at hs
theorem separated_pureCauchy_injective {α : Type*} [UniformSpace α] [s : SeparatedSpace α] :
    @Function.Injective α (Quotient (UniformSpace.separationSetoid (CauchyFilter α)))
      fun a : α => ⟦pureCauchy a⟧
  | a, b, h => by
    refine separated_def.1 s _ _ (fun s hs => ?_)
    rw [← (@uniformEmbedding_pureCauchy α _).comap_uniformity, Filter.mem_comap] at hs
    

Proof state:
α : Type u_1
inst✝ : UniformSpace α
s✝ : SeparatedSpace α
a b : α
h : (fun a => ⟦pureCauchy a⟧) a = (fun a => ⟦pureCauchy a⟧) b
s : Set (α × α)
hs : ∃ t ∈ 𝓤 (CauchyFilter α), (fun x => (pureCauchy x.1, pureCauchy x.2)) ⁻¹' t ⊆ s
⊢ (a, b) ∈ s

Next tactic:
obtain ⟨t, ht, hts⟩ := hs
theorem separated_pureCauchy_injective {α : Type*} [UniformSpace α] [s : SeparatedSpace α] :
    @Function.Injective α (Quotient (UniformSpace.separationSetoid (CauchyFilter α)))
      fun a : α => ⟦pureCauchy a⟧
  | a, b, h => by
    refine separated_def.1 s _ _ (fun s hs => ?_)
    rw [← (@uniformEmbedding_pureCauchy α _).comap_uniformity, Filter.mem_comap] at hs
    obtain ⟨t, ht, hts⟩ := hs
    

Proof state:
case intro.intro
α : Type u_1
inst✝ : UniformSpace α
s✝ : SeparatedSpace α
a b : α
h : (fun a => ⟦pureCauchy a⟧) a = (fun a => ⟦pureCauchy a⟧) b
s : Set (α × α)
t : Set (CauchyFilter α × CauchyFilter α)
ht : t ∈ 𝓤 (CauchyFilter α)
hts : (fun x => (pureCauchy x.1, pureCauchy x.2)) ⁻¹' t ⊆ s
⊢ (a, b) ∈ s

Next tactic:
exact @hts (a, b) (Quotient.exact h t ht)
instance completeSpace_separation [h : CompleteSpace α] :
    CompleteSpace (Quotient (separationSetoid α)) := by
  

Proof state:
α : Type u_1
inst✝² : UniformSpace α
β : Type u_2
inst✝¹ : UniformSpace β
γ : Type u_3
inst✝ : UniformSpace γ
h : CompleteSpace α
⊢ CompleteSpace (Quotient (separationSetoid α))

Next tactic:
constructor
instance completeSpace_separation [h : CompleteSpace α] :
    CompleteSpace (Quotient (separationSetoid α)) := by
  constructor
  

Proof state:
case complete
α : Type u_1
inst✝² : UniformSpace α
β : Type u_2
inst✝¹ : UniformSpace β
γ : Type u_3
inst✝ : UniformSpace γ
h : CompleteSpace α
⊢ ∀ {f : Filter (Quotient (separationSetoid α))}, Cauchy f → ∃ x, f ≤ 𝓝 x

Next tactic:
intro f hf
instance completeSpace_separation [h : CompleteSpace α] :
    CompleteSpace (Quotient (separationSetoid α)) := by
  constructor
  intro f hf
  

Proof state:
case complete
α : Type u_1
inst✝² : UniformSpace α
β : Type u_2
inst✝¹ : UniformSpace β
γ : Type u_3
inst✝ : UniformSpace γ
h : CompleteSpace α
f : Filter (Quotient (separationSetoid α))
hf : Cauchy f
⊢ ∃ x, f ≤ 𝓝 x

Next tactic:
have : Cauchy (f.comap fun x => ⟦x⟧) :=
    hf.comap' comap_quotient_le_uniformity <| hf.left.comap_of_surj (surjective_quotient_mk' _)
instance completeSpace_separation [h : CompleteSpace α] :
    CompleteSpace (Quotient (separationSetoid α)) := by
  constructor
  intro f hf
  have : Cauchy (f.comap fun x => ⟦x⟧) :=
    hf.comap' comap_quotient_le_uniformity <| hf.left.comap_of_surj (surjective_quotient_mk' _)
  

Proof state:
case complete
α : Type u_1
inst✝² : UniformSpace α
β : Type u_2
inst✝¹ : UniformSpace β
γ : Type u_3
inst✝ : UniformSpace γ
h : CompleteSpace α
f : Filter (Quotient (separationSetoid α))
hf : Cauchy f
this : Cauchy (Filter.comap (fun x => ⟦x⟧) f)
⊢ ∃ x, f ≤ 𝓝 x

Next tactic:
let ⟨x, (hx : (f.comap fun x => ⟦x⟧) ≤ 𝓝 x)⟩ := CompleteSpace.complete this
instance completeSpace_separation [h : CompleteSpace α] :
    CompleteSpace (Quotient (separationSetoid α)) := by
  constructor
  intro f hf
  have : Cauchy (f.comap fun x => ⟦x⟧) :=
    hf.comap' comap_quotient_le_uniformity <| hf.left.comap_of_surj (surjective_quotient_mk' _)
  let ⟨x, (hx : (f.comap fun x => ⟦x⟧) ≤ 𝓝 x)⟩ := CompleteSpace.complete this
  

Proof state:
case complete
α : Type u_1
inst✝² : UniformSpace α
β : Type u_2
inst✝¹ : UniformSpace β
γ : Type u_3
inst✝ : UniformSpace γ
h : CompleteSpace α
f : Filter (Quotient (separationSetoid α))
hf : Cauchy f
this : Cauchy (Filter.comap (fun x => ⟦x⟧) f)
x : α
hx : Filter.comap (fun x => ⟦x⟧) f ≤ 𝓝 x
⊢ ∃ x, f ≤ 𝓝 x

Next tactic:
exact ⟨⟦x⟧,
    (comap_le_comap_iff <| by simp).1
      (hx.trans <| map_le_iff_le_comap.1 continuous_quotient_mk'.continuousAt)⟩