awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : (s : SetRel α α) → Set (CauchyFilter α × CauchyFilter α) :=
  fun (s : SetRel α α) ↦ {p : CauchyFilter α × CauchyFilter α | s ∈ (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α)}
h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f
h :
  ∀ (p : CauchyFilter α × CauchyFilter α),
    (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α) = map Prod.swap ((↑p.1 : Filter α) ×ˢ (↑p.2 : Filter α))
simp :
  LE.le (α := Filter (CauchyFilter α × CauchyFilter α))
    ((𝓤 α).lift fun (s : Set (α × α)) ↦
      𝓟 {p : CauchyFilter α × CauchyFilter α | Prod.swap ⁻¹' Prod.swap ⁻¹' s ∈ (↑p.1 : Filter α) ×ˢ (↑p.2 : Filter α)})
    ((𝓤 α).lift' gen)
⊢ LE.le (α := Filter (CauchyFilter α × CauchyFilter α)) ((𝓤 α).lift fun (s : Set (α × α)) ↦ (𝓟 ∘ f) (Prod.swap ⁻¹' s))
    ((𝓤 α).lift' gen)

Tactic:
simp only [Function.comp, h, mem_map, f]

Premises used: Function.comp, Filter.mem_map
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : (s : SetRel α α) → Set (CauchyFilter α × CauchyFilter α) :=
  fun (s : SetRel α α) ↦ {p : CauchyFilter α × CauchyFilter α | s ∈ (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α)}
h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f
h :
  ∀ (p : CauchyFilter α × CauchyFilter α),
    (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α) = map Prod.swap ((↑p.1 : Filter α) ×ˢ (↑p.2 : Filter α))
⊢ LE.le (α := Filter (CauchyFilter α × CauchyFilter α))
    ((𝓤 α).lift fun (s : Set (α × α)) ↦
      𝓟 {p : CauchyFilter α × CauchyFilter α | Prod.swap ⁻¹' Prod.swap ⁻¹' s ∈ (↑p.1 : Filter α) ×ˢ (↑p.2 : Filter α)})
    ((𝓤 α).lift' gen)

Tactic:
le_rfl

Premises used: le_rfl, Set.Mem, setOf, Set.preimage, CauchyFilter.gen, Filter.principal, Prod.swap
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : (s : SetRel α α) → Set (CauchyFilter α × CauchyFilter α) :=
  fun (s : SetRel α α) ↦ {p : CauchyFilter α × CauchyFilter α | s ∈ (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α)}
h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f
h :
  ∀ (p : CauchyFilter α × CauchyFilter α),
    (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α) = map Prod.swap ((↑p.1 : Filter α) ×ˢ (↑p.2 : Filter α))
⊢ LE.le (α := Filter (CauchyFilter α × CauchyFilter α))
    ((𝓤 α).lift fun (s : Set (α × α)) ↦
      𝓟 {p : CauchyFilter α × CauchyFilter α | Prod.swap ⁻¹' Prod.swap ⁻¹' s ∈ (↑p.1 : Filter α) ×ˢ (↑p.2 : Filter α)})
    ((𝓤 α).lift' gen)

Tactic:
exact le_rfl

Premises used: le_rfl
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : (s : SetRel α α) → Set (CauchyFilter α × CauchyFilter α) :=
  fun (s : SetRel α α) ↦ {p : CauchyFilter α × CauchyFilter α | s ∈ (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α)}
h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f
⊢ (𝓤 α).lift' f ≤ (𝓤 α).lift' gen

Tactic:
have h₂ : (𝓤 α).lift' f ≤ (𝓤 α).lift' gen :=
    uniformity_lift_le_swap
      (monotone_principal.comp
        (monotone_setOf fun p => @Filter.monotone_mem _ (p.2.val ×ˢ p.1.val)))
      (by
        have h := fun p : CauchyFilter α × CauchyFilter α => @Filter.prod_comm _ _ p.2.val p.1.val
        simp only [Function.comp, h, mem_map, f]
        exact le_rfl)
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : (s : SetRel α α) → Set (CauchyFilter α × CauchyFilter α) :=
  fun (s : SetRel α α) ↦ {p : CauchyFilter α × CauchyFilter α | s ∈ (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α)}
h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f
h₂ : (𝓤 α).lift' f ≤ (𝓤 α).lift' gen
⊢ LE.le (α := Filter (CauchyFilter α × CauchyFilter α)) (map Prod.swap ((𝓤 α).lift' gen)) ((𝓤 α).lift' gen)

Tactic:
h₁.trans_le h₂

Premises used: Eq.trans_le
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : (s : SetRel α α) → Set (CauchyFilter α × CauchyFilter α) :=
  fun (s : SetRel α α) ↦ {p : CauchyFilter α × CauchyFilter α | s ∈ (↑p.2 : Filter α) ×ˢ (↑p.1 : Filter α)}
h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f
h₂ : (𝓤 α).lift' f ≤ (𝓤 α).lift' gen
⊢ LE.le (α := Filter (CauchyFilter α × CauchyFilter α)) (map Prod.swap ((𝓤 α).lift' gen)) ((𝓤 α).lift' gen)

Tactic:
exact h₁.trans_le h₂

Premises used: Eq.trans_le
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
s t : SetRel α α
⊢ Subset (α := SetRel (CauchyFilter α) (CauchyFilter α)) (gen s ○ gen t) (gen (s ○ t))

Tactic:
fun ⟨f, g⟩ ⟨h, h₁, h₂⟩ =>
  let ⟨t₁, (ht₁ : t₁ ∈ f.val), t₂, (ht₂ : t₂ ∈ h.val), (h₁ : t₁ ×ˢ t₂ ⊆ s)⟩ := mem_prod_iff.mp h₁
  let ⟨t₃, (ht₃ : t₃ ∈ h.val), t₄, (ht₄ : t₄ ∈ g.val), (h₂ : t₃ ×ˢ t₄ ⊆ t)⟩ := mem_prod_iff.mp h₂
  have : t₂ ∩ t₃ ∈ h.val := inter_mem ht₂ ht₃
  let ⟨x, xt₂, xt₃⟩ := h.property.left.nonempty_of_mem this
  (f.val ×ˢ g.val).sets_of_superset (prod_mem_prod ht₁ ht₄)
    fun ⟨a, b⟩ ⟨(ha : a ∈ t₁), (hb : b ∈ t₄)⟩ =>
    ⟨x, h₁ (show (a, x) ∈ t₁ ×ˢ t₂ from ⟨ha, xt₂⟩), h₂ (show (x, b) ∈ t₃ ×ˢ t₄ from ⟨xt₃, hb⟩)⟩

Premises used: _private.0.CauchyFilter.subset_gen_relComp.match_1, SProd.sprod, Filter.inter_mem, _private.0.CauchyFilter.subset_gen_relComp.match_2, Membership.mem, Prod.mk, Filter.sets_of_superset, Set.Subset, SetRel.comp, _private.0.CauchyFilter.subset_gen_relComp.match_4, And.left, And.intro, _private.0.CauchyFilter.subset_gen_relComp.match_6, Exists.intro, Iff.mp, Filter.prod_mem_prod, CauchyFilter.gen, _private.0.CauchyFilter.subset_gen_relComp.match_7, _private.0.CauchyFilter.subset_gen_relComp.match_3, Filter.NeBot.nonempty_of_mem, _private.0.CauchyFilter.subset_gen_relComp.match_5, Subtype.val, Filter.mem_prod_iff, Subtype.property
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
hg : Monotone.{u, u} (α := Set (α × α)) (β := Set (CauchyFilter α × CauchyFilter α)) gen
hh : Monotone (β := Set (CauchyFilter α × CauchyFilter α)) fun (s : Set (CauchyFilter α × CauchyFilter α)) ↦ s ○ s
⊢ Eq (α := Filter (CauchyFilter α × CauchyFilter α))
    (Filter.lift' (α := CauchyFilter α × CauchyFilter α) ((𝓤 α).lift' gen)
      fun (s : Set (CauchyFilter α × CauchyFilter α)) ↦ s ○ s)
    ((𝓤 α).lift' fun (s : Set (α × α)) ↦ gen s ○ gen s)

Tactic:
rw [lift'_lift'_assoc]

Premises used: Filter.lift'_lift'_assoc
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ Monotone.{u, u} (α := Set (α × α)) (β := Set (CauchyFilter α × CauchyFilter α)) gen

Tactic:
monotone_gen

Premises used: CauchyFilter.monotone_gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ Monotone.{u, u} (α := Set (α × α)) (β := Set (CauchyFilter α × CauchyFilter α)) gen

Tactic:
exact monotone_gen

Premises used: CauchyFilter.monotone_gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ Monotone (β := Set (CauchyFilter α × CauchyFilter α)) fun (s : Set (CauchyFilter α × CauchyFilter α)) ↦ s ○ s

Tactic:
monotone_id.relComp monotone_id

Premises used: monotone_id, Monotone.relComp
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ Monotone (β := Set (CauchyFilter α × CauchyFilter α)) fun (s : Set (CauchyFilter α × CauchyFilter α)) ↦ s ○ s

Tactic:
exact monotone_id.relComp monotone_id

Premises used: monotone_id, Monotone.relComp
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
hg : Monotone (β := Set (α × α)) fun (s : SetRel α α) ↦ s ○ s
hh : Monotone.{u, u} (α := Set (α × α)) (β := Set (CauchyFilter α × CauchyFilter α)) gen
⊢ Eq (α := Filter (CauchyFilter α × CauchyFilter α)) ((𝓤 α).lift' fun (s : Set (α × α)) ↦ gen (s ○ s))
    (Filter.lift' ((𝓤 α).lift' fun (s : SetRel α α) ↦ s ○ s) gen)

Tactic:
rw [lift'_lift'_assoc]

Premises used: Filter.lift'_lift'_assoc
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ Monotone (β := Set (α × α)) fun (s : SetRel α α) ↦ s ○ s

Tactic:
monotone_id.relComp monotone_id

Premises used: monotone_id, Monotone.relComp
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ Monotone (β := Set (α × α)) fun (s : SetRel α α) ↦ s ○ s

Tactic:
exact monotone_id.relComp monotone_id

Premises used: monotone_id, Monotone.relComp
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
β : Type v
γ : Type w
inst_1 : UniformSpace β
inst_2 : UniformSpace γ
⊢ Tendsto (α := CauchyFilter α × CauchyFilter α) Prod.swap ((𝓤 α).lift' gen) ((𝓤 α).lift' gen)

Tactic:
symm_gen

Premises used: _private.0.CauchyFilter.symm_gen, Filter.Tendsto
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
β : Type v
γ : Type w
inst_1 : UniformSpace β
inst_2 : UniformSpace γ
⊢ Tendsto (α := CauchyFilter α × CauchyFilter α) Prod.swap ((𝓤 α).lift' gen) ((𝓤 α).lift' gen)

Tactic:
exact symm_gen

Premises used: _private.0.CauchyFilter.symm_gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
β : Type v
γ : Type w
inst_1 : UniformSpace β
inst_2 : UniformSpace γ
⊢ LE.le (α := Filter (CauchyFilter α × CauchyFilter α))
    (Filter.lift' (α := CauchyFilter α × CauchyFilter α) ((𝓤 α).lift' gen)
      fun (s : Set (CauchyFilter α × CauchyFilter α)) ↦ s ○ s)
    ((𝓤 α).lift' gen)

Tactic:
comp_gen

Premises used: _private.0.CauchyFilter.comp_gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
β : Type v
γ : Type w
inst_1 : UniformSpace β
inst_2 : UniformSpace γ
⊢ LE.le (α := Filter (CauchyFilter α × CauchyFilter α))
    (Filter.lift' (α := CauchyFilter α × CauchyFilter α) ((𝓤 α).lift' gen)
      fun (s : Set (CauchyFilter α × CauchyFilter α)) ↦ s ○ s)
    ((𝓤 α).lift' gen)

Tactic:
exact comp_gen

Premises used: _private.0.CauchyFilter.comp_gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
s : Set (CauchyFilter α × CauchyFilter α)
⊢ Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s ↔ ∃ t ∈ 𝓤 α, gen t ⊆ s

Tactic:
mem_lift'_sets monotone_gen

Premises used: Set.Mem, Filter.mem_lift'_sets, CauchyFilter.monotone_gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
ι : Sort u_1
p : ι → Prop
s : ι → SetRel α α
h : (𝓤 α).HasBasis p s
⊢ HasBasis (α := CauchyFilter α × CauchyFilter α) (𝓤 (CauchyFilter α)) p (gen ∘ s)

Tactic:
h.lift' monotone_gen

Premises used: iInf, uniformity, CauchyFilter.monotone_gen, Filter.lift', Filter.lift, Filter.HasBasis.lift'
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
s : Set (CauchyFilter α × CauchyFilter α)
⊢ Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s ↔
    ∃ t ∈ 𝓤 α, ∀ (f g : CauchyFilter α), t ∈ (↑f : Filter α) ×ˢ (↑g : Filter α) → (f, g) ∈ s

Tactic:
mem_uniformity.trans (exists_congr (fun t => and_congr_right_iff.mpr (fun _h => ?_)))

Premises used: Iff.mpr, CauchyFilter.mem_uniformity, and_congr_right_iff, exists_congr, Iff.trans
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
s : Set (CauchyFilter α × CauchyFilter α)
refine :
  ∀ (t : Set (α × α)) (_h : t ∈ 𝓤 α),
    gen t ⊆ s ↔ ∀ (f g : CauchyFilter α), t ∈ (↑f : Filter α) ×ˢ (↑g : Filter α) → (f, g) ∈ s
⊢ Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s ↔
    ∃ t ∈ 𝓤 α, ∀ (f g : CauchyFilter α), t ∈ (↑f : Filter α) ×ˢ (↑g : Filter α) → (f, g) ∈ s

Tactic:
refine mem_uniformity.trans (exists_congr (fun t => and_congr_right_iff.mpr (fun _h => ?_)))

Premises used: Iff.mpr, CauchyFilter.mem_uniformity, and_congr_right_iff, exists_congr, Iff.trans
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
s : Set (CauchyFilter α × CauchyFilter α)
t : Set (α × α)
_h : t ∈ 𝓤 α
⊢ gen t ⊆ s ↔ ∀ (f g : CauchyFilter α), t ∈ (↑f : Filter α) ×ˢ (↑g : Filter α) → (f, g) ∈ s

Tactic:
⟨fun h _f _g ht => h ht, fun h _p hp => h _ _ hp⟩

Premises used: Prod.fst, Iff.intro, Prod.snd
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
s : Set (CauchyFilter α × CauchyFilter α)
t : Set (α × α)
_h : t ∈ 𝓤 α
⊢ gen t ⊆ s ↔ ∀ (f g : CauchyFilter α), t ∈ (↑f : Filter α) ×ˢ (↑g : Filter α) → (f, g) ∈ s

Tactic:
exact ⟨fun h _f _g ht => h ht, fun h _p hp => h _ _ hp⟩
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
s : SetRel α α
x : α × α
a₁ a₂ : α
⊢ Membership.mem (γ := Set (α × α))
      (Function.comp (β := Set (CauchyFilter α × CauchyFilter α))
        (preimage fun (x : α × α) ↦ (pureCauchy x.1, pureCauchy x.2)) gen s)
      (a₁, a₂) ↔
    (a₁, a₂) ∈ id s

Tactic:
simp [preimage, gen, pureCauchy]

Premises used: CauchyFilter.pureCauchy, Set.preimage, CauchyFilter.gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
this :
  Eq.{u + 1} (α := SetRel α α → Set (α × α))
    (Function.comp (β := Set (CauchyFilter α × CauchyFilter α))
      (preimage fun (x : α × α) ↦ (pureCauchy x.1, pureCauchy x.2)) gen)
    id
⊢ (𝓤 α).lift'
      (Function.comp (β := Set (CauchyFilter α × CauchyFilter α))
        (preimage fun (x : α × α) ↦ (pureCauchy x.1, pureCauchy x.2)) gen) =
    𝓤 α

Tactic:
simp [this]
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
s : Set (CauchyFilter α × CauchyFilter α)
hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s
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_1 : t' ∈ (↑f : Filter α) ×ˢ (↑f : Filter α)
t : Set α
ht : t ∈ (↑f : Filter α)
h : t ×ˢ t ⊆ t'
x : α
hx : x ∈ t
this : t'' ∈ SProd.sprod (β := Filter α) (↑f : Filter α) (pure x)
dsimp : t'' ∈ (↑f : Filter α) ×ˢ (↑(pureCauchy x) : Filter α)
⊢ Membership.mem (γ := Set (CauchyFilter α × CauchyFilter α)) (gen t'') (f, pureCauchy x)

Tactic:
dsimp [gen]

Premises used: CauchyFilter.gen
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
s : Set (CauchyFilter α × CauchyFilter α)
hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s
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_1 : t' ∈ (↑f : Filter α) ×ˢ (↑f : Filter α)
t : Set α
ht : t ∈ (↑f : Filter α)
h : t ×ˢ t ⊆ t'
x : α
hx : x ∈ t
this : t'' ∈ SProd.sprod (β := Filter α) (↑f : Filter α) (pure x)
⊢ t'' ∈ (↑f : Filter α) ×ˢ (↑(pureCauchy x) : Filter α)

Tactic:
this

Premises used: Set.Mem, setOf
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
s : Set (CauchyFilter α × CauchyFilter α)
hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s
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_1 : t' ∈ (↑f : Filter α) ×ˢ (↑f : Filter α)
t : Set α
ht : t ∈ (↑f : Filter α)
h : t ×ˢ t ⊆ t'
x : α
hx : x ∈ t
this : t'' ∈ SProd.sprod (β := Filter α) (↑f : Filter α) (pure x)
⊢ t'' ∈ (↑f : Filter α) ×ˢ (↑(pureCauchy x) : Filter α)

Tactic:
exact this
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
⊢ ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s

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 <| SetRel.prodMk_mem_comp (@h (a, x) ⟨h₁, hx⟩) h₂⟩
    ⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
simp :
  NeBot.{u} (α := CauchyFilter α)
    (Filter.lift' (α := CauchyFilter α × CauchyFilter α) (𝓤 (CauchyFilter α))
      fun (t : Set (CauchyFilter α × CauchyFilter α)) ↦ range pureCauchy ∩ UniformSpace.ball f t)
⊢ Membership.mem (γ := Set (CauchyFilter α)) (closure (range pureCauchy)) f

Tactic:
simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
    Set.inter_comm _ (range pureCauchy), mem_setOf_eq]

Premises used: closure_eq_cluster_pts, CauchyFilter.pureCauchy, Set.mem_setOf_eq, Set.inter_comm, ClusterPt, nhds_eq_uniformity, Filter.lift'_inf_principal_eq, Set.range
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
⊢ NeBot.{u} (α := CauchyFilter α)
    (Filter.lift' (α := CauchyFilter α × CauchyFilter α) (𝓤 (CauchyFilter α))
      fun (t : Set (CauchyFilter α × CauchyFilter α)) ↦ range pureCauchy ∩ UniformSpace.ball f t)

Tactic:
(lift'_neBot_iff ?_).mpr (fun s hs => ?_)

Premises used: Iff.mpr, Filter.lift'_neBot_iff
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
refine_1 :
  Monotone (β := Set (CauchyFilter α)) fun (t : Set (CauchyFilter α × CauchyFilter α)) ↦
    range pureCauchy ∩ UniformSpace.ball f t
refine_2 :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α))
    (hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s),
    (range pureCauchy ∩ UniformSpace.ball f s).Nonempty
⊢ NeBot.{u} (α := CauchyFilter α)
    (Filter.lift' (α := CauchyFilter α × CauchyFilter α) (𝓤 (CauchyFilter α))
      fun (t : Set (CauchyFilter α × CauchyFilter α)) ↦ range pureCauchy ∩ UniformSpace.ball f t)

Tactic:
refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)

Premises used: Iff.mpr, Filter.lift'_neBot_iff
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
⊢ Monotone (β := Set (CauchyFilter α)) fun (t : Set (CauchyFilter α × CauchyFilter α)) ↦
    range pureCauchy ∩ UniformSpace.ball f t

Tactic:
monotone_const.inter monotone_preimage

Premises used: monotone_const, Monotone.inter, Set.monotone_preimage
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
⊢ Monotone (β := Set (CauchyFilter α)) fun (t : Set (CauchyFilter α × CauchyFilter α)) ↦
    range pureCauchy ∩ UniformSpace.ball f t

Tactic:
exact monotone_const.inter monotone_preimage

Premises used: monotone_const, Monotone.inter, Set.monotone_preimage
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s
refine_2 : ∀ (y : α) (hy : (f, pureCauchy y) ∈ s), (range pureCauchy ∩ UniformSpace.ball f s).Nonempty
⊢ (range pureCauchy ∩ UniformSpace.ball f s).Nonempty

Tactic:
let ⟨y, hy⟩ := h_ex s hs
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s
y : α
hy : (f, pureCauchy y) ∈ s
⊢ Membership.mem (γ := Set (CauchyFilter α)) (range pureCauchy ∩ {y : CauchyFilter α | (f, y) ∈ s}) (pureCauchy y)

Tactic:
have : pureCauchy y ∈ range pureCauchy ∩ { y : CauchyFilter α | (f, y) ∈ s } :=
      ⟨mem_range_self y, hy⟩
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s
y : α
hy : (f, pureCauchy y) ∈ s
this : Membership.mem (γ := Set (CauchyFilter α)) (range pureCauchy ∩ {y : CauchyFilter α | (f, y) ∈ s}) (pureCauchy y)
⊢ (range pureCauchy ∩ UniformSpace.ball f s).Nonempty

Tactic:
⟨_, this⟩

Premises used: Set.Nonempty, Exists.intro
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
f : CauchyFilter α
h_ex :
  ∀ (s : Set (CauchyFilter α × CauchyFilter α)),
    Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s →
      ∃ (y : α), (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : Membership.mem (γ := Filter (CauchyFilter α × CauchyFilter α)) (𝓤 (CauchyFilter α)) s
y : α
hy : (f, pureCauchy y) ∈ s
this : Membership.mem (γ := Set (CauchyFilter α)) (range pureCauchy ∩ {y : CauchyFilter α | (f, y) ∈ s}) (pureCauchy y)
⊢ (range pureCauchy ∩ UniformSpace.ball f s).Nonempty

Tactic:
exact ⟨_, this⟩
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ IsDenseInducing.{u, u} (α := α) (β := CauchyFilter α) pureCauchy

Tactic:
isUniformInducing_pureCauchy.isDenseInducing denseRange_pureCauchy

Premises used: CauchyFilter.isUniformInducing_pureCauchy, IsUniformInducing.isDenseInducing, CauchyFilter.denseRange_pureCauchy
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
⊢ IsDenseEmbedding.{u, u} (α := α) (β := CauchyFilter α) pureCauchy

Tactic:
isUniformEmbedding_pureCauchy.isDenseEmbedding denseRange_pureCauchy

Premises used: IsUniformEmbedding.isDenseEmbedding, CauchyFilter.denseRange_pureCauchy, CauchyFilter.isUniformEmbedding_pureCauchy
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
mp : Nonempty.{u + 1} (CauchyFilter α) → Nonempty α
mpr : Nonempty α → Nonempty.{u + 1} (CauchyFilter α)
⊢ Nonempty.{u + 1} (CauchyFilter α) ↔ Nonempty α

Tactic:
constructor
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
c : CauchyFilter α
⊢ Membership.mem (γ := Set (CauchyFilter α)) (closure (range pureCauchy)) c

Tactic:
have := eq_univ_iff_forall.1 isDenseEmbedding_pureCauchy.isDenseInducing.closure_range c
-- Module: Mathlib.Topology.UniformSpace.Completion
Goal:
α : Type u
inst : UniformSpace α
c : CauchyFilter α
this : Membership.mem (γ := Set (CauchyFilter α)) (closure (range pureCauchy)) c
mp :
  ∀ (w : CauchyFilter α), Membership.mem (γ := Set (CauchyFilter α)) univ w → ∀ (a : α), pureCauchy a = w → Nonempty α
⊢ Nonempty α

Tactic:
obtain ⟨_, ⟨_, a, _⟩⟩ := mem_closure_iff.1 this _ isOpen_univ trivial

Premises used: mem_closure_iff, trivial, isOpen_univ, Iff.mp