awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
inst_1 : SeparableSpace α
s : Set α
hsc : s.Countable
hsd : Dense s
t : ℕ → SetRel α α
h_basis : (𝓤 α).HasAntitoneBasis t
ht_mem : ∀ (i : ℕ), t i ∈ (𝓤 α).sets
hto : ∀ (i : ℕ), IsOpen[instTopologicalSpaceProd] (t i)
hts : ∀ (i : ℕ), (t i).IsSymm
x : α
V : Set α
hxV : x ∈ V
hVo : IsOpen[uniformSpace.toTopologicalSpace] V
U : Set (α × α)
hU : U ∈ 𝓤 α
hUV : ball x U ⊆ V
U' : Set (α × α)
hU' : U' ∈ 𝓤 α
left : ∀ {a b : α}, (a, b) ∈ U' → (b, a) ∈ U'
hUU' : U' ○ U' ⊆ U
k : ℕ
hk : t k ⊆ U'
⊢ ∃ (v : Set α), (∃ i ∈ s, ∃ (y : ℕ), ball i (t y) = v) ∧ x ∈ v ∧ v ⊆ V
have refine_2 : ∀ (y : α) (hxy : y ∈ ball x (t k)) (hys : y ∈ s), ∃ (v : Set α), (∃ i ∈ s, ∃ (y : ℕ), ball i (t y) = v) ∧ x ∈ v ∧ v ⊆ V
Tactic: rcases hsd.inter_open_nonempty (ball x <| t k) (isOpen_ball x (hto k))
        ⟨x, UniformSpace.mem_ball_self _ (ht_mem k)⟩ with
      ⟨y, hxy, hys⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
inst_1 : SeparableSpace α
s : Set α
hsc : s.Countable
hsd : Dense s
t : ℕ → SetRel α α
h_basis : (𝓤 α).HasAntitoneBasis t
ht_mem : ∀ (i : ℕ), t i ∈ (𝓤 α).sets
hto : ∀ (i : ℕ), IsOpen[instTopologicalSpaceProd] (t i)
hts : ∀ (i : ℕ), (t i).IsSymm
x : α
V : Set α
hxV : x ∈ V
hVo : IsOpen[uniformSpace.toTopologicalSpace] V
U : Set (α × α)
hU : U ∈ 𝓤 α
hUV : ball x U ⊆ V
U' : Set (α × α)
hU' : U' ∈ 𝓤 α
left : ∀ {a b : α}, (a, b) ∈ U' → (b, a) ∈ U'
hUU' : U' ○ U' ⊆ U
k : ℕ
hk : t k ⊆ U'
y : α
hxy : y ∈ ball x (t k)
hys : y ∈ s
⊢ ∃ (v : Set α), (∃ i ∈ s, ∃ (y : ℕ), ball i (t y) = v) ∧ x ∈ v ∧ v ⊆ V
have refine_2 : ∀ (z : α) (hz : z ∈ ball y (t k)), z ∈ V
Tactic: refine ⟨_, ⟨y, hys, k, rfl⟩, (t k).symm hxy, fun z hz => ?_⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
hs : ∀ U ∈ 𝓤 α, ∃ (t : Set α), t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
have obtain : ∀ (B : ℕ → SetRel α α) (hB : (𝓤 α).HasAntitoneBasis B),
  (∀ (n : ℕ), (B n).IsSymm) → ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
Tactic: obtain ⟨B, hB, _⟩ := has_seq_basis α
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (hypothesis)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
hs : ∀ U ∈ 𝓤 α, ∃ (t : Set α), t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
have hs₁ : ∀ (n : ℕ), ∃ (t : Set α), t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x (B n)
Tactic: replace hs (n : ℕ) := hs (B n) (hB.mem n)
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
hs : ∀ (n : ℕ), ∃ (t : Set α), t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x (B n)
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
have choose : ∀ (t : (n : ℕ) → Set α) (tC : ∀ (n : ℕ), (t n).Countable) (ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)),
  ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
Tactic: choose t tC ht using hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (hypothesis)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
have this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
Tactic: have := fun n => (tC n).to_subtype
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
have choose : ∀ (o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α)
  (hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n))
  (hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s),
  ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
Tactic: choose o hox hos using fun (n : ℕ) (x : t n) (hx : (ball x.1 (B n) ∩ s).Nonempty) => hx
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure[uniformSpace.toTopologicalSpace] t
have refine : ∀ (x : α) (hx : x ∈ s),
  Membership.mem (γ := Set α)
    (closure[uniformSpace.toTopologicalSpace] (⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x))) x
Tactic: refine ⟨⋃ (n) (x), range (o n x), iUnion₂_subset fun _ _ => range_subset_iff.2 (hos _ _),
    countable_iUnion fun _ => countable_iUnion fun _ => countable_range _, fun x hx => ?_⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
⊢ Membership.mem (γ := Set α)
    (closure[uniformSpace.toTopologicalSpace] (⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x))) x
have rw : ∀ {V : Set (α × α)}, V ∈ 𝓤 α → (ball x V ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
Tactic: rw [mem_closure_iff_ball]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
have obtain : ∀ (V : Set (α × α)) (hV : V ∈ 𝓤 α) (hVU : V ○ V ⊆ U),
  (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
Tactic: obtain ⟨V, hV, hVU⟩ := comp_mem_uniformity_sets hU
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : V ○ V ⊆ U
⊢ (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
have obtain : ∀ (n : ℕ) (hn : B n ⊆ V), (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
Tactic: obtain ⟨n, hn⟩ := hB.mem_iff.1 hV
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (hypothesis)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
ht : ∀ (n : ℕ), s ⊆ ⋃ x ∈ t n, ball x (B n)
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : V ○ V ⊆ U
n : ℕ
hn : B n ⊆ V
⊢ (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
have ht₁ : Membership.mem (γ := Set α) (⋃ x ∈ t n, ball x (B n)) x
Tactic: specialize ht n hx
Cleared hypotheses: ht
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (hypothesis)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : V ○ V ⊆ U
n : ℕ
hn : B n ⊆ V
ht : Membership.mem (γ := Set α) (⋃ x ∈ t n, ball x (B n)) x
⊢ (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
have ht₁ : ∃ (i : α) (_ : i ∈ t n), x ∈ ball i (B n)
Tactic: rw [mem_iUnion₂] at ht
Cleared hypotheses: ht
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : V ○ V ⊆ U
n : ℕ
hn : B n ⊆ V
ht : ∃ (i : α) (_ : i ∈ t n), x ∈ ball i (B n)
⊢ (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
have obtain : ∀ (y : α) (hy : y ∈ t n) (hyx : x ∈ ball y (B n)),
  (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
Tactic: obtain ⟨y, hy, hyx⟩ := ht
Cleared hypotheses: ht
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : V ○ V ⊆ U
n : ℕ
hn : B n ⊆ V
y : α
hy : y ∈ t n
hyx : x ∈ ball y (B n)
⊢ (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
have refine_1 : o n (Subtype.mk (p := fun (x : α) ↦ x ∈ t n) y hy) sorry ∈ ball x U
Tactic: refine ⟨o n ⟨y, hy⟩ ⟨x, hyx, hx⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : V ○ V ⊆ U
n : ℕ
hn : B n ⊆ V
y : α
hy : y ∈ t n
hyx : x ∈ ball y (B n)
refine_1 : o n (Subtype.mk (p := fun (x : α) ↦ x ∈ t n) y hy) ⋯ ∈ ball x U
⊢ (ball x U ∩ ⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x)).Nonempty
have refine_2 : Membership.mem (γ := Set α) (⋃ (n : ℕ), ⋃ (x : (↑(t n) : Type u)), range (o n x))
  (o n (Subtype.mk (p := fun (x : α) ↦ x ∈ t n) y hy) sorry)
Tactic: refine ⟨o n ⟨y, hy⟩ ⟨x, hyx, hx⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
B : ℕ → SetRel α α
hB : (𝓤 α).HasAntitoneBasis B
right : ∀ (n : ℕ), (B n).IsSymm
t : (n : ℕ) → Set α
tC : ∀ (n : ℕ), (t n).Countable
this : ∀ (n : ℕ), Countable.{u + 1} (↑(t n) : Type u)
o : (n : ℕ) → (x : (↑(t n) : Type u)) → (hx : (ball (↑x : α) (B n) ∩ s).Nonempty) → α
hox : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ ball (↑x : α) (B n)
hos : ∀ (n : ℕ) (x : (↑(t n) : Type u)) (hx : (ball (↑x : α) (B n) ∩ s).Nonempty), o n x hx ∈ s
x : α
hx : x ∈ s
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : V ○ V ⊆ U
n : ℕ
hn : B n ⊆ V
y : α
hy : y ∈ t n
hyx : x ∈ ball y (B n)
⊢ o n (Subtype.mk (p := fun (x : α) ↦ x ∈ t n) y hy) ⋯ ∈ ball x U
have refine_1 : o n (Subtype.mk (p := fun (x : α) ↦ x ∈ t n) y hy) sorry ∈ ball x (B n ○ B n)
Tactic: apply ball_mono ((SetRel.comp_subset_comp hn hn).trans hVU)
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
hs : ∀ U ∈ 𝓤 α, ∃ (t : Set α), t.Countable ∧ Eq.{u + 1} (α := Set α) (⋃ x ∈ t, ball x U) univ
⊢ SecondCountableTopology α
have suffices : SeparableSpace α
Tactic: suffices SeparableSpace α from UniformSpace.secondCountable_of_separable α
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (hypothesis)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
hs : ∀ U ∈ 𝓤 α, ∃ (t : Set α), t.Countable ∧ Eq.{u + 1} (α := Set α) (⋃ x ∈ t, ball x U) univ
⊢ SeparableSpace α
have this : ∀ U ∈ 𝓤 α, ∃ (t : Set α), t.Countable ∧ Subset.{u} (α := Set α) univ (⋃ x ∈ t, ball x U)
Tactic: have : ∀ U ∈ 𝓤 α, ∃ t : Set α, Set.Countable t ∧ univ ⊆ ⋃ x ∈ t, ball x U := by
    simpa only [univ_subset_iff] using hs
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
hs : ∀ U ∈ 𝓤 α, ∃ (t : Set α), t.Countable ∧ Eq.{u + 1} (α := Set α) (⋃ x ∈ t, ball x U) univ
this : ∀ U ∈ 𝓤 α, ∃ (t : Set α), t.Countable ∧ Subset.{u} (α := Set α) univ (⋃ x ∈ t, ball x U)
⊢ SeparableSpace α
have rcases : ∀ (t : Set α) (htc : t.Countable) (ht : univ ⊆ closure[uniformSpace.toTopologicalSpace] t), SeparableSpace α
Tactic: rcases subset_countable_closure_of_almost_dense_set (univ : Set α) this with ⟨t, -, htc, ht⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
h : TotallyBounded s
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∃ (t : Set α), t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U
have obtain : ∀ (t : Set α) (ht : t.Finite) (hst : s ⊆ ⋃ y ∈ t, {x : α | (x, y) ∈ SetRel.inv U}),
  ∃ (t : Set α), t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U
Tactic: obtain ⟨t, ht, hst⟩ := h (SetRel.inv U)
      (mem_of_superset (symmetrize_mem_uniformity hU) SetRel.symmetrize_subset_inv)
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Cauchy (goal)
Goal:
α : Type u
uniformSpace : UniformSpace α
inst : (𝓤 α).IsCountablyGenerated
s : Set α
h : TotallyBounded s
⊢ IsSeparable s
have obtain : ∀ (t : Set α) (htc : t.Countable) (hts : s ⊆ closure[uniformSpace.toTopologicalSpace] t), IsSeparable s
Tactic: obtain ⟨t, -, htc, hts⟩ := subset_countable_closure_of_almost_dense_set s fun U hU => by
    obtain ⟨t, ht, hst⟩ := h (SetRel.inv U)
      (mem_of_superset (symmetrize_mem_uniformity hU) SetRel.symmetrize_subset_inv)
    exact ⟨t, ht.countable, hst⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
⊢ Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) (hausdorffEntourage SetRel.id) SetRel.id
have simp_rw : Eq.{u_1 + 1} (α := SetRel (Set α) (Set α))
  {x : Set α × Set α | x.1 ⊆ preimage SetRel.id x.2 ∧ x.2 ⊆ image SetRel.id x.1} SetRel.id
Tactic: simp_rw [hausdorffEntourage, preimage_id, image_id, ← subset_antisymm_iff, SetRel.id]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
simp_rw :
  Eq.{u_1 + 1} (α := SetRel (Set α) (Set α))
    {x : Set α × Set α | x.1 ⊆ preimage SetRel.id x.2 ∧ x.2 ⊆ image SetRel.id x.1} SetRel.id
⊢ Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) (hausdorffEntourage SetRel.id) SetRel.id
have simp_rw₁ : Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) {x : Set α × Set α | x.1 ⊆ x.2 ∧ x.2 ⊆ image SetRel.id x.1} SetRel.id
Tactic: simp_rw [hausdorffEntourage, preimage_id, image_id, ← subset_antisymm_iff, SetRel.id]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
simp_rw :
  Eq.{u_1 + 1} (α := SetRel (Set α) (Set α))
    {x : Set α × Set α | x.1 ⊆ preimage SetRel.id x.2 ∧ x.2 ⊆ image SetRel.id x.1} SetRel.id
simp_rw₁ :
  Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) {x : Set α × Set α | x.1 ⊆ x.2 ∧ x.2 ⊆ image SetRel.id x.1} SetRel.id
⊢ Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) (hausdorffEntourage SetRel.id) SetRel.id
have simp_rw₂ : Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) {x : Set α × Set α | x.1 ⊆ x.2 ∧ x.2 ⊆ x.1} SetRel.id
Tactic: simp_rw [hausdorffEntourage, preimage_id, image_id, ← subset_antisymm_iff, SetRel.id]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
simp_rw :
  Eq.{u_1 + 1} (α := SetRel (Set α) (Set α))
    {x : Set α × Set α | x.1 ⊆ preimage SetRel.id x.2 ∧ x.2 ⊆ image SetRel.id x.1} SetRel.id
simp_rw₁ :
  Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) {x : Set α × Set α | x.1 ⊆ x.2 ∧ x.2 ⊆ image SetRel.id x.1} SetRel.id
simp_rw₂ : Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) {x : Set α × Set α | x.1 ⊆ x.2 ∧ x.2 ⊆ x.1} SetRel.id
⊢ Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) (hausdorffEntourage SetRel.id) SetRel.id
have simp_rw₃ : Eq.{u_1 + 1} (α := SetRel (Set α) (Set α)) {x : Set α × Set α | x.1 = x.2} SetRel.id
Tactic: simp_rw [hausdorffEntourage, preimage_id, image_id, ← subset_antisymm_iff, SetRel.id]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst : U.IsSymm
⊢ (hausdorffEntourage U).IsSymm
have rw : (hausdorffEntourage U).inv = hausdorffEntourage U
Tactic: rw [← inv_eq_self_iff, inv_hausdorffEntourage, inv_eq_self]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst : U.IsSymm
rw : (hausdorffEntourage U).inv = hausdorffEntourage U
⊢ (hausdorffEntourage U).IsSymm
have rw₁ : hausdorffEntourage U.inv = hausdorffEntourage U
Tactic: rw [← inv_eq_self_iff, inv_hausdorffEntourage, inv_eq_self]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst : U.IsSymm
rw : (hausdorffEntourage U).inv = hausdorffEntourage U
rw₁ : hausdorffEntourage U.inv = hausdorffEntourage U
⊢ (hausdorffEntourage U).IsSymm
have rw₂ : hausdorffEntourage U = hausdorffEntourage U
Tactic: rw [← inv_eq_self_iff, inv_hausdorffEntourage, inv_eq_self]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
U V : SetRel α α
⊢ hausdorffEntourage (U ○ V) = hausdorffEntourage U ○ hausdorffEntourage V
have a : hausdorffEntourage (U ○ V) ⊆ hausdorffEntourage U ○ hausdorffEntourage V
Tactic: apply subset_antisymm
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
U V : SetRel α α
a : hausdorffEntourage (U ○ V) ⊆ hausdorffEntourage U ○ hausdorffEntourage V
⊢ hausdorffEntourage (U ○ V) = hausdorffEntourage U ○ hausdorffEntourage V
have a₁ : hausdorffEntourage U ○ hausdorffEntourage V ⊆ hausdorffEntourage (U ○ V)
Tactic: apply subset_antisymm
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
U V : SetRel α α
s t : Set α
⊢ (s, t) ∈ hausdorffEntourage (U ○ V) → (s, t) ∈ hausdorffEntourage U ○ hausdorffEntourage V
have a : ∀ (hst : (s, t).1 ⊆ (U ○ V).preimage (s, t).2) (hts : (s, t).2 ⊆ (U ○ V).image (s, t).1),
  (s, t) ∈ hausdorffEntourage U ○ hausdorffEntourage V
Tactic: ⟨hst, hts⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
U V : SetRel α α
s t : Set α
hst : (s, t).1 ⊆ (U ○ V).preimage (s, t).2
hts : (s, t).2 ⊆ (U ○ V).image (s, t).1
⊢ (s, t) ∈ hausdorffEntourage U ○ hausdorffEntourage V
have a : ∃ (b : Set α), (s ⊆ U.preimage b ∧ b ⊆ U.image s) ∧ b ⊆ V.preimage t ∧ t ⊆ V.image b
Tactic: simp only [mem_comp, mem_hausdorffEntourage] at *
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
⊢ ∃ (b : Set α), (s ⊆ U.preimage b ∧ b ⊆ U.image s) ∧ b ⊆ V.preimage t ∧ t ⊆ V.image b
have a : s ⊆ U.preimage (U.image s ∩ V.preimage t)
Tactic: refine ⟨U.image s ∩ V.preimage t, ⟨?_, Set.inter_subset_left⟩, ⟨Set.inter_subset_right, ?_⟩⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
a : s ⊆ U.preimage (U.image s ∩ V.preimage t)
⊢ ∃ (b : Set α), (s ⊆ U.preimage b ∧ b ⊆ U.image s) ∧ b ⊆ V.preimage t ∧ t ⊆ V.image b
have a₁ : t ⊆ V.image (U.image s ∩ V.preimage t)
Tactic: refine ⟨U.image s ∩ V.preimage t, ⟨?_, Set.inter_subset_left⟩, ⟨Set.inter_subset_right, ?_⟩⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
x : α
hx : x ∈ s
⊢ x ∈ U.preimage (U.image s ∩ V.preimage t)
have a : ∀ (z : α) (hz : z ∈ t) (y : α) (hxy : (x, y) ∈ U) (hyz : (y, z) ∈ V), x ∈ U.preimage (U.image s ∩ V.preimage t)
Tactic: obtain ⟨z, hz, y, hxy, hyz⟩ := hst hx