l3lab/ntp-mathlib huggingface dataset:
private theorem comp_gen : (((𝓤 α).lift' gen).lift' fun s => compRel s s) ≤ (𝓤 α).lift' gen :=
  calc
    (((𝓤 α).lift' gen).lift' fun s => compRel s s) =
        (𝓤 α).lift' fun s => compRel (gen s) (gen s) := by
      

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ (Filter.lift' (Filter.lift' (𝓤 α) gen) fun s => s ○ s) = Filter.lift' (𝓤 α) fun s => gen s ○ gen s

Next tactic:
rw [lift'_lift'_assoc]
private theorem comp_gen : (((𝓤 α).lift' gen).lift' fun s => compRel s s) ≤ (𝓤 α).lift' gen :=
  calc
    (((𝓤 α).lift' gen).lift' fun s => compRel s s) =
        (𝓤 α).lift' fun s => compRel (gen s) (gen s) := by
      rw [lift'_lift'_assoc]
      · 

Proof state:
case hg
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ Monotone gen

Next tactic:
exact monotone_gen
private theorem comp_gen : (((𝓤 α).lift' gen).lift' fun s => compRel s s) ≤ (𝓤 α).lift' gen :=
  calc
    (((𝓤 α).lift' gen).lift' fun s => compRel s s) =
        (𝓤 α).lift' fun s => compRel (gen s) (gen s) := by
      rw [lift'_lift'_assoc]
      · exact monotone_gen
      · 

Proof state:
case hh
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ Monotone fun s => s ○ s

Next tactic:
exact monotone_id.compRel monotone_id
private theorem comp_gen : (((𝓤 α).lift' gen).lift' fun s => compRel s s) ≤ (𝓤 α).lift' gen :=
  calc
    (((𝓤 α).lift' gen).lift' fun s => compRel s s) =
        (𝓤 α).lift' fun s => compRel (gen s) (gen s) := by
      rw [lift'_lift'_assoc]
      · exact monotone_gen
      · exact monotone_id.compRel monotone_id
    _ ≤ (𝓤 α).lift' fun s => gen <| compRel s s :=
      lift'_mono' fun s _hs => compRel_gen_gen_subset_gen_compRel
    _ = ((𝓤 α).lift' fun s : Set (α × α) => compRel s s).lift' gen := by
      

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ (Filter.lift' (𝓤 α) fun s => gen (s ○ s)) = Filter.lift' (Filter.lift' (𝓤 α) fun s => s ○ s) gen

Next tactic:
rw [lift'_lift'_assoc]
private theorem comp_gen : (((𝓤 α).lift' gen).lift' fun s => compRel s s) ≤ (𝓤 α).lift' gen :=
  calc
    (((𝓤 α).lift' gen).lift' fun s => compRel s s) =
        (𝓤 α).lift' fun s => compRel (gen s) (gen s) := by
      rw [lift'_lift'_assoc]
      · exact monotone_gen
      · exact monotone_id.compRel monotone_id
    _ ≤ (𝓤 α).lift' fun s => gen <| compRel s s :=
      lift'_mono' fun s _hs => compRel_gen_gen_subset_gen_compRel
    _ = ((𝓤 α).lift' fun s : Set (α × α) => compRel s s).lift' gen := by
      rw [lift'_lift'_assoc]
      · 

Proof state:
case hg
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ Monotone fun s => s ○ s

Next tactic:
exact monotone_id.compRel monotone_id
private theorem comp_gen : (((𝓤 α).lift' gen).lift' fun s => compRel s s) ≤ (𝓤 α).lift' gen :=
  calc
    (((𝓤 α).lift' gen).lift' fun s => compRel s s) =
        (𝓤 α).lift' fun s => compRel (gen s) (gen s) := by
      rw [lift'_lift'_assoc]
      · exact monotone_gen
      · exact monotone_id.compRel monotone_id
    _ ≤ (𝓤 α).lift' fun s => gen <| compRel s s :=
      lift'_mono' fun s _hs => compRel_gen_gen_subset_gen_compRel
    _ = ((𝓤 α).lift' fun s : Set (α × α) => compRel s s).lift' gen := by
      rw [lift'_lift'_assoc]
      · exact monotone_id.compRel monotone_id
      · 

Proof state:
case hh
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ Monotone gen

Next tactic:
exact monotone_gen
theorem mem_uniformity' {s : Set (CauchyFilter α × CauchyFilter α)} :
    s ∈ 𝓤 (CauchyFilter α) ↔ ∃ t ∈ 𝓤 α, ∀ f g : CauchyFilter α, t ∈ f.1 ×ˢ g.1 → (f, g) ∈ s := by
  

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
s : Set (CauchyFilter α × CauchyFilter α)
⊢ s ∈ 𝓤 (CauchyFilter α) ↔ ∃ t ∈ 𝓤 α, ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s

Next tactic:
refine mem_uniformity.trans (exists_congr (fun t => and_congr_right_iff.mpr (fun _h => ?_)))
theorem mem_uniformity' {s : Set (CauchyFilter α × CauchyFilter α)} :
    s ∈ 𝓤 (CauchyFilter α) ↔ ∃ t ∈ 𝓤 α, ∀ f g : CauchyFilter α, t ∈ f.1 ×ˢ g.1 → (f, g) ∈ s := by
  refine mem_uniformity.trans (exists_congr (fun t => and_congr_right_iff.mpr (fun _h => ?_)))
  

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
s : Set (CauchyFilter α × CauchyFilter α)
t : Set (α × α)
_h : t ∈ 𝓤 α
⊢ gen t ⊆ s ↔ ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s

Next tactic:
exact ⟨fun h _f _g ht => h ht, fun h _p hp => h _ _ hp⟩
theorem uniformInducing_pureCauchy : UniformInducing (pureCauchy : α → CauchyFilter α) :=
  ⟨have : (preimage fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen = id :=
      funext fun s =>
        Set.ext fun ⟨a₁, a₂⟩ => by 

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
s : Set (α × α)
x✝ : α × α
a₁ a₂ : α
⊢ (a₁, a₂) ∈ ((preimage fun x => (pureCauchy x.1, pureCauchy x.2)) ∘ gen) s ↔ (a₁, a₂) ∈ id s

Next tactic:
simp [preimage, gen, pureCauchy, prod_principal_principal]
theorem uniformInducing_pureCauchy : UniformInducing (pureCauchy : α → CauchyFilter α) :=
  ⟨have : (preimage fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen = id :=
      funext fun s =>
        Set.ext fun ⟨a₁, a₂⟩ => by simp [preimage, gen, pureCauchy, prod_principal_principal]
    calc
      comap (fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ((𝓤 α).lift' gen) =
          (𝓤 α).lift' ((preimage fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen) :=
        comap_lift'_eq
      _ = 𝓤 α := by 

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
this : (preimage fun x => (pureCauchy x.1, pureCauchy x.2)) ∘ gen = id
⊢ Filter.lift' (𝓤 α) ((preimage fun x => (pureCauchy x.1, pureCauchy x.2)) ∘ gen) = 𝓤 α

Next tactic:
simp [this]
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun 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 : CauchyFilter α
⊢ f ∈ closure (range pureCauchy)

Next tactic:
have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by 

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ 𝓤 (CauchyFilter α)
t'' : Set (α × α)
ht''₁ : t'' ∈ 𝓤 α
ht''₂ : gen t'' ⊆ s
t' : Set (α × α)
ht'₁ : t' ∈ 𝓤 α
ht'₂ : t' ○ t' ⊆ t''
this✝ : t' ∈ ↑f ×ˢ ↑f
t : Set α
ht : t ∈ ↑f
h : t ×ˢ t ⊆ t'
x : α
hx : x ∈ t
this : t'' ∈ ↑f ×ˢ pure x
⊢ (f, pureCauchy x) ∈ gen t''

Next tactic:
dsimp [gen]
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; 

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ 𝓤 (CauchyFilter α)
t'' : Set (α × α)
ht''₁ : t'' ∈ 𝓤 α
ht''₂ : gen t'' ⊆ s
t' : Set (α × α)
ht'₁ : t' ∈ 𝓤 α
ht'₂ : t' ○ t' ⊆ t''
this✝ : t' ∈ ↑f ×ˢ ↑f
t : Set α
ht : t ∈ ↑f
h : t ×ˢ t ⊆ t'
x : α
hx : x ∈ t
this : t'' ∈ ↑f ×ˢ pure x
⊢ t'' ∈ ↑f ×ˢ ↑(pureCauchy x)

Next tactic:
exact this
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
⊢ f ∈ closure (range pureCauchy)

Next tactic:
simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
  

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
⊢ NeBot (Filter.lift' (𝓤 (CauchyFilter α)) fun t => range pureCauchy ∩ UniformSpace.ball f t)

Next tactic:
refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
  refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
  · 

Proof state:
case refine_1
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
⊢ Monotone fun t => range pureCauchy ∩ UniformSpace.ball f t

Next tactic:
refine monotone_const.inter ?_
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
  refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
  · refine monotone_const.inter ?_
    

Proof state:
case refine_1
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
⊢ Monotone fun t => UniformSpace.ball f t

Next tactic:
simp_rw [UniformSpace.ball]
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
  refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
  · refine monotone_const.inter ?_
    simp_rw [UniformSpace.ball]
    

Proof state:
case refine_1
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
⊢ Monotone fun t => Prod.mk f ⁻¹' t

Next tactic:
exact monotone_preimage
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
  refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
  · refine monotone_const.inter ?_
    simp_rw [UniformSpace.ball]
    exact monotone_preimage
  · 

Proof state:
case refine_2
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ 𝓤 (CauchyFilter α)
⊢ Set.Nonempty (range pureCauchy ∩ UniformSpace.ball f s)

Next tactic:
let ⟨y, hy⟩ := h_ex s hs
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
  refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
  · refine monotone_const.inter ?_
    simp_rw [UniformSpace.ball]
    exact monotone_preimage
  · let ⟨y, hy⟩ := h_ex s hs
    

Proof state:
case refine_2
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ 𝓤 (CauchyFilter α)
y : α
hy : (f, pureCauchy y) ∈ s
⊢ Set.Nonempty (range pureCauchy ∩ UniformSpace.ball f s)

Next tactic:
have : pureCauchy y ∈ range pureCauchy ∩ { y : CauchyFilter α | (f, y) ∈ s } :=
      ⟨mem_range_self y, hy⟩
theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
  have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
    let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
    let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
    have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
    let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
    let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
    have : t'' ∈ f.val ×ˢ pure x :=
      mem_prod_iff.mpr
        ⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
          fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
          ht'₂ <| prod_mk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
  simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
  refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
  · refine monotone_const.inter ?_
    simp_rw [UniformSpace.ball]
    exact monotone_preimage
  · let ⟨y, hy⟩ := h_ex s hs
    have : pureCauchy y ∈ range pureCauchy ∩ { y : CauchyFilter α | (f, y) ∈ s } :=
      ⟨mem_range_self y, hy⟩
    

Proof state:
case refine_2
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ 𝓤 (CauchyFilter α)
y : α
hy : (f, pureCauchy y) ∈ s
this : pureCauchy y ∈ range pureCauchy ∩ {y | (f, y) ∈ s}
⊢ Set.Nonempty (range pureCauchy ∩ UniformSpace.ball f s)

Next tactic:
exact ⟨_, this⟩
theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
  

Proof state:
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ Nonempty (CauchyFilter α) ↔ Nonempty α

Next tactic:
constructor
theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
  constructor <;> 

Proof state:
case mp
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ Nonempty (CauchyFilter α) → Nonempty α

Next tactic:
rintro ⟨c⟩
theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
  constructor <;> 

Proof state:
case mpr
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ Nonempty α → Nonempty (CauchyFilter α)

Next tactic:
rintro ⟨c⟩
theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
  constructor <;> rintro ⟨c⟩
  · 

Proof state:
case mp.intro
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
c : CauchyFilter α
⊢ Nonempty α

Next tactic:
have := eq_univ_iff_forall.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c
theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
  constructor <;> rintro ⟨c⟩
  · have := eq_univ_iff_forall.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c
    

Proof state:
case mp.intro
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
c : CauchyFilter α
this : c ∈ closure (range pureCauchy)
⊢ Nonempty α

Next tactic:
obtain ⟨_, ⟨_, a, _⟩⟩ := mem_closure_iff.1 this _ isOpen_univ trivial
theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
  constructor <;> rintro ⟨c⟩
  · have := eq_univ_iff_forall.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c
    obtain ⟨_, ⟨_, a, _⟩⟩ := mem_closure_iff.1 this _ isOpen_univ trivial
    

Proof state:
case mp.intro.intro.intro.intro
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
c : CauchyFilter α
this : c ∈ closure (range pureCauchy)
w✝ : CauchyFilter α
left✝ : w✝ ∈ univ
a : α
h✝ : pureCauchy a = w✝
⊢ Nonempty α

Next tactic:
exact ⟨a⟩
theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
  constructor <;> rintro ⟨c⟩
  · have := eq_univ_iff_forall.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c
    obtain ⟨_, ⟨_, a, _⟩⟩ := mem_closure_iff.1 this _ isOpen_univ trivial
    exact ⟨a⟩
  · 

Proof state:
case mpr.intro
α : Type u
inst✝² : UniformSpace α
β : Type v
γ : Type w
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
c : α
⊢ Nonempty (CauchyFilter α)

Next tactic:
exact ⟨pureCauchy c⟩