l3lab/ntp-mathlib huggingface dataset:
theorem totallyBounded_of_forall_symm {s : Set α}
    (h : ∀ V ∈ 𝓤 α, SymmetricRel V → ∃ t : Set α, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y V) :
    TotallyBounded s :=
  UniformSpace.hasBasis_symmetric.totallyBounded_iff.2 fun V hV => by
    

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
h : ∀ V ∈ 𝓤 α, SymmetricRel V → ∃ t, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y V
V : Set (α × α)
hV : V ∈ 𝓤 α ∧ SymmetricRel V
⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ id V}

Next tactic:
simpa only [ball_eq_of_symmetry hV.2] using h V hV.1 hV.2
/-- The image of a totally bounded set under a uniformly continuous map is totally bounded. -/
theorem TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
    (hf : UniformContinuous f) : TotallyBounded (f '' s) := fun t ht =>
  have : { p : α × α | (f p.1, f p.2) ∈ t } ∈ 𝓤 α := hf ht
  let ⟨c, hfc, hct⟩ := hs _ this
  ⟨f '' c, hfc.image f, by
    

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
hs : TotallyBounded s
hf : UniformContinuous f
t : Set (β × β)
ht : t ∈ 𝓤 β
this : {p | (f p.1, f p.2) ∈ t} ∈ 𝓤 α
c : Set α
hfc : Set.Finite c
hct : s ⊆ ⋃ y ∈ c, {x | (x, y) ∈ {p | (f p.1, f p.2) ∈ t}}
⊢ f '' s ⊆ ⋃ y ∈ f '' c, {x | (x, y) ∈ t}

Next tactic:
simp only [mem_image, iUnion_exists, biUnion_and', iUnion_iUnion_eq_right, image_subset_iff,
      preimage_iUnion, preimage_setOf_eq]
/-- The image of a totally bounded set under a uniformly continuous map is totally bounded. -/
theorem TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
    (hf : UniformContinuous f) : TotallyBounded (f '' s) := fun t ht =>
  have : { p : α × α | (f p.1, f p.2) ∈ t } ∈ 𝓤 α := hf ht
  let ⟨c, hfc, hct⟩ := hs _ this
  ⟨f '' c, hfc.image f, by
    simp only [mem_image, iUnion_exists, biUnion_and', iUnion_iUnion_eq_right, image_subset_iff,
      preimage_iUnion, preimage_setOf_eq]
    

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
hs : TotallyBounded s
hf : UniformContinuous f
t : Set (β × β)
ht : t ∈ 𝓤 β
this : {p | (f p.1, f p.2) ∈ t} ∈ 𝓤 α
c : Set α
hfc : Set.Finite c
hct : s ⊆ ⋃ y ∈ c, {x | (x, y) ∈ {p | (f p.1, f p.2) ∈ t}}
⊢ s ⊆ ⋃ i ∈ c, {a | (f a, f i) ∈ t}

Next tactic:
simp? [subset_def] at hct says
      simp only [mem_setOf_eq, subset_def, mem_iUnion, exists_prop] at hct
/-- The image of a totally bounded set under a uniformly continuous map is totally bounded. -/
theorem TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
    (hf : UniformContinuous f) : TotallyBounded (f '' s) := fun t ht =>
  have : { p : α × α | (f p.1, f p.2) ∈ t } ∈ 𝓤 α := hf ht
  let ⟨c, hfc, hct⟩ := hs _ this
  ⟨f '' c, hfc.image f, by
    simp only [mem_image, iUnion_exists, biUnion_and', iUnion_iUnion_eq_right, image_subset_iff,
      preimage_iUnion, preimage_setOf_eq]
    simp? [subset_def] at hct says
      

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
hs : TotallyBounded s
hf : UniformContinuous f
t : Set (β × β)
ht : t ∈ 𝓤 β
this : {p | (f p.1, f p.2) ∈ t} ∈ 𝓤 α
c : Set α
hfc : Set.Finite c
hct : s ⊆ ⋃ y ∈ c, {x | (x, y) ∈ {p | (f p.1, f p.2) ∈ t}}
⊢ s ⊆ ⋃ i ∈ c, {a | (f a, f i) ∈ t}

Next tactic:
simp only [mem_setOf_eq, subset_def, mem_iUnion, exists_prop] at hct
/-- The image of a totally bounded set under a uniformly continuous map is totally bounded. -/
theorem TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
    (hf : UniformContinuous f) : TotallyBounded (f '' s) := fun t ht =>
  have : { p : α × α | (f p.1, f p.2) ∈ t } ∈ 𝓤 α := hf ht
  let ⟨c, hfc, hct⟩ := hs _ this
  ⟨f '' c, hfc.image f, by
    simp only [mem_image, iUnion_exists, biUnion_and', iUnion_iUnion_eq_right, image_subset_iff,
      preimage_iUnion, preimage_setOf_eq]
    simp? [subset_def] at hct says
      simp only [mem_setOf_eq, subset_def, mem_iUnion, exists_prop] at hct
    

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
hs : TotallyBounded s
hf : UniformContinuous f
t : Set (β × β)
ht : t ∈ 𝓤 β
this : {p | (f p.1, f p.2) ∈ t} ∈ 𝓤 α
c : Set α
hfc : Set.Finite c
hct : ∀ x ∈ s, ∃ i ∈ c, (f x, f i) ∈ t
⊢ s ⊆ ⋃ i ∈ c, {a | (f a, f i) ∈ t}

Next tactic:
intro x hx
/-- The image of a totally bounded set under a uniformly continuous map is totally bounded. -/
theorem TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
    (hf : UniformContinuous f) : TotallyBounded (f '' s) := fun t ht =>
  have : { p : α × α | (f p.1, f p.2) ∈ t } ∈ 𝓤 α := hf ht
  let ⟨c, hfc, hct⟩ := hs _ this
  ⟨f '' c, hfc.image f, by
    simp only [mem_image, iUnion_exists, biUnion_and', iUnion_iUnion_eq_right, image_subset_iff,
      preimage_iUnion, preimage_setOf_eq]
    simp? [subset_def] at hct says
      simp only [mem_setOf_eq, subset_def, mem_iUnion, exists_prop] at hct
    intro x hx; 

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
hs : TotallyBounded s
hf : UniformContinuous f
t : Set (β × β)
ht : t ∈ 𝓤 β
this : {p | (f p.1, f p.2) ∈ t} ∈ 𝓤 α
c : Set α
hfc : Set.Finite c
hct : ∀ x ∈ s, ∃ i ∈ c, (f x, f i) ∈ t
x : α
hx : x ∈ s
⊢ x ∈ ⋃ i ∈ c, {a | (f a, f i) ∈ t}

Next tactic:
simp
/-- The image of a totally bounded set under a uniformly continuous map is totally bounded. -/
theorem TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
    (hf : UniformContinuous f) : TotallyBounded (f '' s) := fun t ht =>
  have : { p : α × α | (f p.1, f p.2) ∈ t } ∈ 𝓤 α := hf ht
  let ⟨c, hfc, hct⟩ := hs _ this
  ⟨f '' c, hfc.image f, by
    simp only [mem_image, iUnion_exists, biUnion_and', iUnion_iUnion_eq_right, image_subset_iff,
      preimage_iUnion, preimage_setOf_eq]
    simp? [subset_def] at hct says
      simp only [mem_setOf_eq, subset_def, mem_iUnion, exists_prop] at hct
    intro x hx; simp
    

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
hs : TotallyBounded s
hf : UniformContinuous f
t : Set (β × β)
ht : t ∈ 𝓤 β
this : {p | (f p.1, f p.2) ∈ t} ∈ 𝓤 α
c : Set α
hfc : Set.Finite c
hct : ∀ x ∈ s, ∃ i ∈ c, (f x, f i) ∈ t
x : α
hx : x ∈ s
⊢ ∃ i ∈ c, (f x, f i) ∈ t

Next tactic:
exact hct x hx
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
⊢ TotallyBounded s ↔ ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c

Next tactic:
constructor
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · 

Proof state:
case mp
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
⊢ TotallyBounded s → ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c

Next tactic:
exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · 

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
⊢ (∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c) → TotallyBounded s

Next tactic:
intro H d hd
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
H : ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c
d : Set (α × α)
hd : d ∈ 𝓤 α
⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}

Next tactic:
contrapose! H with hd_cover
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ c ≤ f, ¬Cauchy c

Next tactic:
set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ c ≤ f, ¬Cauchy c

Next tactic:
have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ c ≤ f, ¬Cauchy c

Next tactic:
have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this : NeBot f
⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ c ≤ f, ¬Cauchy c

Next tactic:
have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by 

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this : NeBot f
⊢ 𝓟 (s \ ⋃ y ∈ ∅, {x | (x, y) ∈ d}) ≤ 𝓟 s

Next tactic:
simp
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
    

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝ : NeBot f
this : f ≤ 𝓟 s
⊢ ∃ f, NeBot f ∧ f ≤ 𝓟 s ∧ ∀ c ≤ f, ¬Cauchy c

Next tactic:
refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
    refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩
    

Proof state:
case mpr
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝ : NeBot f
this : f ≤ 𝓟 s
c : Filter α
hcf : c ≤ f
hc : Cauchy c
⊢ False

Next tactic:
rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
    refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩
    rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩
    

Proof state:
case mpr.intro.intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝ : NeBot f
this : f ≤ 𝓟 s
c : Filter α
hcf : c ≤ f
hc : Cauchy c
m : Set α
hm : m ∈ c
hmd : m ×ˢ m ⊆ d
⊢ False

Next tactic:
rcases hc.1.nonempty_of_mem hm with ⟨y, hym⟩
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
    refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩
    rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩
    rcases hc.1.nonempty_of_mem hm with ⟨y, hym⟩
    

Proof state:
case mpr.intro.intro.intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝ : NeBot f
this : f ≤ 𝓟 s
c : Filter α
hcf : c ≤ f
hc : Cauchy c
m : Set α
hm : m ∈ c
hmd : m ×ˢ m ⊆ d
y : α
hym : y ∈ m
⊢ False

Next tactic:
have : s \ {x | (x, y) ∈ d} ∈ c := by simpa using hcf (hb.mem {y})
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
    refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩
    rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩
    rcases hc.1.nonempty_of_mem hm with ⟨y, hym⟩
    have : s \ {x | (x, y) ∈ d} ∈ c := by 

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝ : NeBot f
this : f ≤ 𝓟 s
c : Filter α
hcf : c ≤ f
hc : Cauchy c
m : Set α
hm : m ∈ c
hmd : m ×ˢ m ⊆ d
y : α
hym : y ∈ m
⊢ s \ {x | (x, y) ∈ d} ∈ c

Next tactic:
simpa using hcf (hb.mem {y})
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
    refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩
    rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩
    rcases hc.1.nonempty_of_mem hm with ⟨y, hym⟩
    have : s \ {x | (x, y) ∈ d} ∈ c := by simpa using hcf (hb.mem {y})
    

Proof state:
case mpr.intro.intro.intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝¹ : NeBot f
this✝ : f ≤ 𝓟 s
c : Filter α
hcf : c ≤ f
hc : Cauchy c
m : Set α
hm : m ∈ c
hmd : m ×ˢ m ⊆ d
y : α
hym : y ∈ m
this : s \ {x | (x, y) ∈ d} ∈ c
⊢ False

Next tactic:
rcases hc.1.nonempty_of_mem (inter_mem hm this) with ⟨z, hzm, -, hyz⟩
theorem totallyBounded_iff_filter {s : Set α} :
    TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
  constructor
  · exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
      (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
  · intro H d hd
    contrapose! H with hd_cover
    set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
    have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
      .iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
    have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
      nonempty_diff.2 <| hd_cover _ (Finset.finite_toSet _)
    have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
    refine' ⟨f, ‹_›, ‹_›, fun c hcf hc => _⟩
    rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩
    rcases hc.1.nonempty_of_mem hm with ⟨y, hym⟩
    have : s \ {x | (x, y) ∈ d} ∈ c := by simpa using hcf (hb.mem {y})
    rcases hc.1.nonempty_of_mem (inter_mem hm this) with ⟨z, hzm, -, hyz⟩
    

Proof state:
case mpr.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
d : Set (α × α)
hd : d ∈ 𝓤 α
hd_cover : ∀ (t : Set α), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ d}
f : Filter α := ⨅ t, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d})
hb : HasAntitoneBasis f fun t => s \ ⋃ y ∈ t, {x | (x, y) ∈ d}
this✝¹ : NeBot f
this✝ : f ≤ 𝓟 s
c : Filter α
hcf : c ≤ f
hc : Cauchy c
m : Set α
hm : m ∈ c
hmd : m ×ˢ m ⊆ d
y : α
hym : y ∈ m
this : s \ {x | (x, y) ∈ d} ∈ c
z : α
hzm : z ∈ m
hyz : z ∉ {x | (x, y) ∈ d}
⊢ False

Next tactic:
exact hyz (hmd ⟨hzm, hym⟩)
theorem totallyBounded_iff_ultrafilter {s : Set α} :
    TotallyBounded s ↔ ∀ f : Ultrafilter α, ↑f ≤ 𝓟 s → Cauchy (f : Filter α) := by
  

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
⊢ TotallyBounded s ↔ ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f

Next tactic:
refine' ⟨fun hs f => f.cauchy_of_totallyBounded hs, fun H => totallyBounded_iff_filter.2 _⟩
theorem totallyBounded_iff_ultrafilter {s : Set α} :
    TotallyBounded s ↔ ∀ f : Ultrafilter α, ↑f ≤ 𝓟 s → Cauchy (f : Filter α) := by
  refine' ⟨fun hs f => f.cauchy_of_totallyBounded hs, fun H => totallyBounded_iff_filter.2 _⟩
  

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
H : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f
⊢ ∀ (f : Filter α), NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c

Next tactic:
intro f hf hfs
theorem totallyBounded_iff_ultrafilter {s : Set α} :
    TotallyBounded s ↔ ∀ f : Ultrafilter α, ↑f ≤ 𝓟 s → Cauchy (f : Filter α) := by
  refine' ⟨fun hs f => f.cauchy_of_totallyBounded hs, fun H => totallyBounded_iff_filter.2 _⟩
  intro f hf hfs
  

Proof state:
α : Type u
β : Type v
uniformSpace : UniformSpace α
s : Set α
H : ∀ (f : Ultrafilter α), ↑f ≤ 𝓟 s → Cauchy ↑f
f : Filter α
hf : NeBot f
hfs : f ≤ 𝓟 s
⊢ ∃ c ≤ f, Cauchy c

Next tactic:
exact ⟨Ultrafilter.of f, Ultrafilter.of_le f, H _ ((Ultrafilter.of_le f).trans hfs)⟩